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f93db7e215a76cbeb868c6a5bafcf09a2d0066ac | subsection | 203 | 224 | Integral formulae for commutators | In this section of the appendix, we collect results concerning formulae for commutators with functions of D. Many of the results of this section
will be known to the expert reader, but since they are scattered around various sources we provide them here with short and self-contained proofs.In this section, (\mathcal {A... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.04301958158612251,
0.004786309786140919,
-0.021830148994922638,
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0.013912715017795563,
0.006159274838864803,
0.033652905374765396,
0.031303610652685165,
-0.... | |
822b3315c7af09923049879ae9f7ee95497ed34e | subsection | 204 | 224 | Integral formulae for commutators | Therefore in particular,
this is an integral in the weak operator topology.We can compute the terms in the integrand as:\xi ^{\prime }(v)\eta (v) &= -itp_n\exp (it(1-v)|D|)|D|x\exp (itv|D|)p_n,\\
\xi (v)\eta ^{\prime }(v) &= itp_n\exp (it(1-v)|D|)x|D|\exp (itv|D|)p_n.Thus,\xi (1)\eta (1)-\xi (0)\eta (0) = -it\int _0^1 ... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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0.014723316766321659,
0.00574... | |
0feb1f0bf12bd8c90c7dc7c302ee802c50988471 | subsection | 205 | 224 | Integral formulae for commutators | The following formula is well known and appears in many places, for example .Lemma 6.2.2
If \widehat{f},\widehat{f^{\prime }}\in L_1(\mathbb {R}), then for all x \in \mathcal {B}, and s > 0,[f(s|D|),x] = s\int _{-\infty }^{\infty }\left(\int _0^1\widehat{f^{\prime }}(u)e^{ius(1-v)|D|}\delta (x)e^{iusv|D|}dv\right)\,du... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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0.024001743644475937,
0... | |
dae52d7d522df210446082bf32a07ba54000d0c9 | subsection | 206 | 224 | Integral formulae for commutators | Performing a linear change of variables, w_0 = vw, we get:\int _0^1 ve^{ius(1-vw)|D|}\delta ^2(x)e^{iusvw|D|}\,dw = \int _0^v e^{ius(1-w_0)|D|}\delta ^2(x)e^{iusw_0|D|}\,dw_0,and therefore:\int _0^1 \int _0^1 ve^{ius(1-vw)|D|}\delta ^2(x)e^{iusvw|D|}\,dwdv = \int _0^1 \int _{0}^v e^{ius(1-w)|D|}\delta ^2(x)e^{iusw|D|}\... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.049687858670949936,
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0.03096335008740425,
-0.0188... | |
90664c7e6eca52b2ab0f71bdb7f0f7b162035cb8 | subsection | 207 | 224 | Hochschild coboundary computations | In this part of the appendix we include some of the lengthy algebraic computations required for Sections and .
Recall that for a multilinear functional \theta :\mathcal {A}^{\otimes p}\rightarrow \mathbb {C} the Hochschild coboundary b\theta :\mathcal {A}^{\otimes (p+1)}\rightarrow \mathbb {C} is defined in terms of th... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.03536557778716087,
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-0.006823664531111717,
-0.011297760531306267,
-0.0... | |
74f4210fc899af8b22208a459b0bc065f396a046 | subsection | 208 | 224 | Coboundaries in Section | Let {A} \subseteq \lbrace 1,\ldots ,p\rbrace . Let T :=D^{2-|{A}|}|D|^{p+1}e^{-s^2D^2}.
Following the notation of Definition REF , we define
for a \in \mathcal {A},b_k(a) := {\left\lbrace \begin{array}{ll}
\delta (a),\quad k \in {A},\\
[F,a],\quad k\notin {A}.
\end{array}\right.}Fix 1\le m\le p-1. We introduce a pair o... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.0417582131922245,
-0.023199006915092468,
-0.06349201500415802,
0.010798221454024315,
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0.027838807553052902,
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0.010775327682495117,
0.006528536323457956,
-0.005... | |
2979ccfe6f522ed959998b13aa63be7a7ad634af | subsection | 209 | 224 | Coboundaries in Section | For 1\le k < m we also introduce X_k^1 and X_k^2 defined by:X_k^1 &:= {\rm Tr}\left(\Gamma a_0\left(\prod _{l=1}^{k-1}b_l(a_l)\right)a_k\left(\prod _{l=k}^{m-2}b_l(a_{l+1})\right)\delta ^2(a_m)\left(\prod _{l=m+1}^pb_l(a_l)\right)\cdot T\right),\\
X_k^2 &:= {\rm Tr}\left(\Gamma a_0\left(\prod _{l=1}^{k-1}b_l(a_l)\right... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.0527065135538578,
-0.014824660494923592,
-0.04608386382460594,
0.017121223732829094,
0.005310325883328915,
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0.015229038894176483,
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-0.025651317089796066,
-0.029542503878474236,
-0.... | |
9f104f48814cccf17991135f1ff8a1e9a309e397 | subsection | 210 | 224 | Coboundaries in Section | By the definition of \theta _s^j, we have\theta _s^1(a_0\otimes &a_1\otimes \cdots \otimes a_{k-1}\otimes a_{k}a_{k+1}\otimes a_{k+2}\otimes \cdots \otimes a_p)\\
&= {\rm Tr}\left(\Gamma a_0\left(\prod _{l=1}^{k-1}b_l(a_l)\right)b_k(a_ka_{k+1})\left(\prod _{l=k+1}^{m-2}b_l(a_{l+1})\right)\delta ^2(a_m)\left(\prod _{l=m... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.0584663487970829,
0.04204938933253288,
-0.04842698201537132,
0.016828909516334534,
-0.007052732165902853,
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-0.003944037482142448,
0.04711484536528587,
0.013601969927549362,
0.025830773636698723,
-0.05010529235005379,
-0.012038086540997028,
-0.02206219732761383,
0.00... | |
480e0819efb8cb03bb2402e63b52ba405893d3c6 | subsection | 211 | 224 | Coboundaries in Section | By definition we have:\theta _s^1(&a_0\otimes a_1\otimes \cdots \otimes a_{k-1}\otimes a_{k}a_{k+1}\otimes a_{k+2}\otimes \cdots \otimes a_p)\\
&= {\rm Tr}\left(\Gamma a_0\left(\prod _{l=1}^{m-2}b_l(a_l)\right)\delta ^2(a_{m-1})\left(\prod _{l=m}^{k-1}b_{l+1}(a_l)\right)b_{k+1}(a_ka_{k+1})\left(\prod _{l=m+1}^{p-1}b_{l... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.03562094271183014,
0.029073650017380714,
-0.054972853511571884,
0.015261758118867874,
-0.005089796148240566,
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0.044625382870435715,
0.02826477587223053,
0.017993612214922905,
-0.04831872507929802,
0.0012743568513542414,
-0.007703372277319431,
... | |
675923f11a9e48fff783efcbc824c4b32061f269 | subsection | 212 | 224 | Coboundaries in Section | If we assume that m-1,m \in {A}, then we have:\theta _s^1(&a_0\otimes a_1\otimes \cdots \otimes a_{m-2}\otimes a_{m-1}a_m\otimes a_{m+1}\otimes \cdots \otimes a_p)\\
&=X_{m-1}^1+Y_m^1+2{\rm Tr}(\mathcal {W}_{{A}}(c)\cdot T).Now if {B} \subseteq \lbrace 1,\ldots , p\rbrace is such that |{B}|=|{A}| and {A}\Delta {B} = \l... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.0700729489326477,
0.036013100296258926,
-0.05108977109193802,
0.007973238825798035,
-0.004734348971396685,
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0.030565354973077774,
0.017182521522045135,
-0.04709170386195183,
-0.0258500799536705,
-0.02292020060122013,
0.0022... | |
ebd34630321967f8d2536f809bd54ee51ee43b1b | subsection | 213 | 224 | Coboundaries in Section | Since \Gamma commutes with a_p, we have:\theta _s^1(&a_pa_0\otimes a_1\otimes a_2\otimes \cdots \otimes a_{p-1})\\
&= {\rm Tr}\left(\Gamma a_pa_0\left(\prod _{l=1}^{m-2}b_l(a_l)\right)\delta ^2(a_{m-1})\left(\prod _{l=m}^{p-1}b_{l+1}(a_l)\right)\cdot T\right)\\
&= {\rm Tr}\left(a_p\Gamma a_0\left(\prod _{l=1}^{m-2}b_l(... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.045656342059373856,
0.01139882579445839,
-0.03665325045585632,
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0.025651171803474426,
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0.02047821134328842,
0.010208587162196636,
-0.013... | |
7d3f31314539a212f2c4e1d89a19d2f9e1eb5b69 | subsection | 214 | 224 | Coboundaries in Section | Let T = Fe^{-s^2D^2}. | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.05220408737659454,
0.036490991711616516,
-0.051563359797000885,
0.0037337669637054205,
-0.013317991979420185,
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0.0043706814758479595,
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-0.0045003523118793964,
-0.0157436057925224... | |
fd965c03feff77d9422f016afbfc474fda258afb | subsection | 215 | 224 | Coboundaries in Section | Note that this is different
to T in the preceding section.We define the multilinear mapping \mathcal {L}_s:\mathcal {A}^{\otimes p}\rightarrow \mathbb {C} on a_0\otimes \cdots \otimes a_{p-1} \in \mathcal {A}^{\otimes p} by:\mathcal {L}_s(a_0\otimes \cdots \otimes a_{p-1}) := {\rm Tr}\left(\Gamma a_0\left(\prod _{k=1}^... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.04239197447896004,
0.0021974241826683283,
-0.0571330264210701,
-0.01529803965240717,
0.017213154584169388,
0.01782355085015297,
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0.0402555875480175,
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0.002983308397233486,
-0.04294132813811302,
0.0011111106723546982,
0.014641864225268364,
0.00... | |
bfed89e92da5d56452669e759567f7010d59a598 | subsection | 216 | 224 | Coboundaries in Section | We define the multilinear mapping \theta _s:\mathcal {A}^{\otimes p}\rightarrow \mathbb {C} on a_0\otimes \cdots a_{p-1} \in \mathcal {A}^{\otimes p} by:\theta _s(a_0\otimes \cdots \otimes a_{p-1}) = {\rm Tr}\left(\left(\prod _{k=0}^{p-1}\partial (a_k)\right) T\right).For 0 \le k \le p we also define:X_k = {\rm Tr}\lef... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.03948933258652687,
-0.0012130610411986709,
-0.04235795512795448,
0.0234677717089653,
-0.001688939519226551,
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0.013336041942238808,
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-0.009300134144723415,
0.000... | |
f3f893487a4d092d0ae4080a1e8a6bb21eaeee58 | subsection | 217 | 224 | Technical estimates for Section 4.4.1 | For this section, (\mathcal {A},H,D) is assumed to be a spectral triple satisfying Hypothesis REF and we
assume that D has a spectral gap at 0.The results of this section are very similar to that of Lemma REF . However additional technicalities make the proofs more involved and therefore are included here in the append... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.08559826761484146,
0.00348773249424994,
-0.025612320750951767,
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0.006494661793112755,
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0.016164151951670647,
0.045760273933410645,
-0... | |
468de81cffaf5d4b1669e8b89fd11dcc2ee4638a | subsection | 218 | 224 | Technical estimates for Section 4.4.1 | Then on H_\infty :|D|^{-n}&\left(\prod _{k=1}^mb_k(a_k)\right)|D|^{n+m-|{B}|}\\
&=\left(|D|^{-n}\left(\prod _{k=1}^{m-1}b_k(a_k)\right)|D|^{n+(m-1)-|{C}|}\right)\left(|D|^{-n_1}b_m(a_m)|D|^{n_1+1-|{B}\cap \lbrace m\rbrace |}\right).By the inductive assumption, the first factor has bounded extension.If m\in {B}, then th... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.059159521013498306,
0.030342519283294678,
-0.016948485746979713,
0.002982384292408824,
-0.025109998881816864,
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0.0015159818576648831,
0.05900697037577629,
0.025521887466311455,
0.010937951505184174,
-0.06050197407603264,
-0.0032836738973855972,
-0.0014044283889234066,... | |
09cc93a1921cd6367bf2356863bc158c2b64577e | subsection | 219 | 224 | Technical estimates for Section 4.4.1 | On H_\infty we have\Gamma a_0\left(\prod _{k=1}^{m-2}b_k(a_k)\right)\delta ^2(a_{m-1})\left(\prod _{k=m}^{p-1}b_{k+1}(a_k)\right)|D|^{p-|{A}|}= \mathrm {I}\cdot \mathrm {II}\cdot \mathrm {III}.Here,\mathrm {I} &= \Gamma a_0\left(\prod _{k=1}^{m-2}b_k(a_k)\right)|D|^{n_1},\\
\mathrm {II} &= |D|^{-n_1}\delta ^2(a_{m-1})|... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.0943591445684433,
0.022048668935894966,
-0.021682463586330414,
-0.01426678616553545,
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-0.0010146965505555272,
-0.0008773691952228546,
-... | |
ce4fe0a9a8ad9e05cc2a495b13179c5ed08687a8 | subsection | 220 | 224 | Technical estimates for Section 4.4.1 | We have\Gamma a_0\left(\prod _{k=1}^{m-2}b_k(a_k)\right)[F,\delta (a_{m-1})]\left(\prod _{k=m}^{p-1}b_{k+1}(a_k)\right)|D|^{p-|{A}|}=\Gamma a_0\cdot \mathrm {I}\cdot \mathrm {II}\cdot \mathrm {III}.Here,\mathrm {I} &= \left(\prod _{k=1}^{m-2}b_k(a_k)\right)|D|^{n_1},\\
\mathrm {II} &= |D|^{-n_1}[F,\delta (a_{m-1})]|D|^... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.07837404310703278,
0.02711656503379345,
-0.028047408908605576,
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0.028734097257256508,
0.008186852559447289,
-0.06836365163326263,
0.001171185402199626,
-0.011902600526809692,
... | |
82580ffea8fb4eeed436973705ae6309041b43eb | subsection | 221 | 224 | Subkhankulov's computation | The following assertion is identical to . However to the best of our knowledge there is no published proof in English,
and is not easily accessible. For the convenience of the reader we include a proof here.Proposition 6.5.1
For all u\in (0,1) and v\in \mathbb {R}, we have\frac{1}{2\pi }\int _{-1}^1\frac{(1-t^2)^2}{u+... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1112/jlms/s2-32.1.116",
"end": 41,
"openalex_id": "https://openalex.org/W2132508118",
"raw": "Subhankulov M. Tauberian theorems with remainders. Nauka, Moscow, 1976.",
"source_ref_id": "7f1e88aaed3dcde8c983d893122744bd8f3df486",
... | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.052509840577840805,
0.019527193158864975,
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ccb30561a27b6737da2e02fe08d1189b459cca4f | subsection | 222 | 224 | Subkhankulov's computation | By construction,
the point -u is in the interior of the curve \gamma and so
by the Cauchy integral formula we have:\frac{1}{2\pi i}\int _{\gamma }f(z)e^{zv}dz = (1+u^2)^2e^{-uv}.Since \gamma = \gamma _0\cup \gamma _1:\frac{1}{2\pi i}\int _{\gamma _0}f(z)e^{zv}dz+\frac{1}{2\pi i}\int _{\gamma _1}f(z)e^{zv}dz = (1+u^2)^2... | {
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} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
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477b65a1b08cfa93ab7d978e77b94795eab75e82 | subsection | 223 | 224 | Subkhankulov's computation | This proof is similar, but instead we consider a contour in the half plane \lbrace z\;:\;\Re (z)\ge 0\rbrace .
Let \gamma _2 be a smooth curve without self-intersections such that\gamma _2 starts at -i and ends at i.
\gamma _2 lies in the half-plane \lbrace \Re (z)\ge 0\rbrace .
the distance between \gamma _2 and [-1... | {
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} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
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a3b249ce0078de6927f4914a15e61d5613e63efe | abstract | 0 | 89 | Abstract | We determine the group of reductive cohomological degree $3$ invariants of
all split semisimple groups of types $B$, $C$, and $D$. We also present a
complete description of the cohomological invariants. As an application, we
show that the group of degree $3$ unramified cohomology of the classifying
space $BG$ is trivia... | {
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} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
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91f6c4546a7abfc6489520df8e5e8228d8236eeb | subsection | 1 | 89 | Introduction | A degree d cohomological invariant of an algebraic group G defined over a field F is a natural transformation of functorsG\operatorname{-{torsors}}\rightarrow H^{d}on the category of field extensions over F, where the functor G\operatorname{-{torsors}} takes a field K/F to the set G\operatorname{-{torsors}}(K) of isomo... | {
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... | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
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da57116f1b822a3371ac48a09860ba3d527d2083 | subsection | 2 | 89 | Introduction | Recently, this subgroup has been completely computed for all split simple groups in and for all split semisimple groups of type A in .In the present paper, we determine the group of reductive indecomposable invariants of all split semisimple groups of types B, C, and D, which completes the cohomological invariants of c... | {
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27072e617b98292412cd52cc999f57858d0952dd | subsection | 3 | 89 | Introduction | In particular, if n_{i}\ge 2 for all 1\le i\le m, then\operatorname{Inv}^{3}(G)_{\operatorname{\hspace{0.85358pt}ind}}=\operatorname{Inv}^{3}(G)_{\operatorname{red}}=(\mathbb {Z}/2\mathbb {Z})^{l-l_{1}}.(2) Assume that G is of type C. Let s denote the number of ranks n_{i} divisible by 4 and l=\dim \big (R\cap (\bigopl... | {
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} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
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2b1998767aee53a982980955e63d28f3d29ed844 | subsection | 4 | 89 | Introduction | Set&R^{\prime }=\bar{R}\cap \big (\bigoplus _{4\nmid n_{i}, R_{1,i}^{\prime }, R_{1,i}\ne Z_{i}}(\mathbb {Z}/2\mathbb {Z})\bar{e}_{i}\big ) \text{ with } l=\dim R^{\prime }, \, I_{1}=\lbrace i\,|\, Z_{i}=R_{1,i} \text{ or } R_{1,i}^{\prime }, n_{i}\ne 3\rbrace ,\\
&I_{2}=\lbrace i \,|\,R_{1,i}^{\prime }=0, 4|n_{i}\rbra... | {
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0596a7cff9e6247db4e5e1650ad553487f224533 | subsection | 5 | 89 | Introduction | Then, this morphism is surjective and its kernel is the subspace\langle e_{i},\, e_{j}+e_{k}\in R\,|\, e_{j}, e_{k}\notin R,\, n_{i}\le 2,\, n_{j}=n_{k}=1\rangle .For type C, let G_{\operatorname{red}}=(\prod _{i=1}^{m}\operatorname{\mathbf {GSp}}_{2n_{i}})/{\mu }, where \operatorname{\mathbf {GSp}}_{2n_{i}} is the gro... | {
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2484494a45dcb591c4c51ce82164f3628c4a2824 | subsection | 6 | 89 | Introduction | Then, this morphism is surjective and its kernel is given by\langle e_{i},\, e_{j}+e_{k}\in R\,|\, e_{j}, e_{k}\notin R,\, n_{j}\equiv n_{k}\equiv 1\mod {2}\rangle .For type D, let G_{\operatorname{red}}=(\prod _{i=1}^{m}\operatorname{\mathbf {\Omega }}_{2n_{i}})/{\mu }, where \operatorname{\mathbf {\Omega }}_{2n_{i}} ... | {
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5d7386b7bc7bef455b7e9f056b8d97051544aed3 | subsection | 7 | 89 | Introduction | Then, the morphism is surjective, and its kernel is given by\langle \bar{e}_{i},\, \bar{e}_{j}+\bar{e}_{k}\in R^{\prime }\,|\, \bar{e}_{j}, \bar{e}_{k}\notin R^{\prime },\, n_{j}\equiv n_{k}\equiv 1\mod {2}\rangle .Therefore, our main result (Theorem REF ) tells us that for all split semisimple groups of types B, C, D ... | {
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7d827568b062d87e7da3fa51cb32856566fcd1c6 | subsection | 8 | 89 | Introduction | Recently, Merkurjev has shown that the group \operatorname{Inv}^{3}_{\operatorname{nr}}(G) is trivial if G is a split simple group or a split semisimple group of type A over an algebraically field F of characteristic 0.Using the main theorem above we determine the group of unramified invariants of a split semisimple gr... | {
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99087c123e6b5acb971dc005d99656ddf5a35c04 | subsection | 9 | 89 | Body | I am grateful to Alexander Merkurjev for careful reading and numerous suggestions. I am also grateful to Jean-Pierre Tignol for helpful discussion. This work has been supported by National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (2016R1C1B2010037). | {
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} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
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a301b98978cf3060ea6bf48f43c37a6c6efaf3c2 | subsection | 10 | 89 | Cohomological invariants of degree | In this section we recall some basic notions concerning degree 3 invariants following , . We shall frequently use these in the following sections. | {
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35de5a1016ab7bc00ee947f01cdb984af6be0d27 | subsection | 11 | 89 | Invariant quadratic forms | Let \tilde{G} be a split semisimple simply connected group of Dynkin type \mathcal {D}, i.e., \tilde{G}=G_{1}\times \cdots \times G_{m} for some integer m\ge 1, where each G_{i} is a split simple simply connected group of type \mathcal {D}. Consider the natural action of the Weyl group W=W_{1}\times \cdots W_{m} of \ti... | {
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} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
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03efc47464e58cb4e99c2d27305322bb2398134b | subsection | 12 | 89 | Degree | Consider the Chern class map c_{2}:\mathbb {Z}[T^{*}]\rightarrow S^{2}(T^{*}) defined by c_{2}(\sum _{i}e^{\lambda _{i}})=\sum _{i<j}\lambda _{i}\lambda _{j} , where \mathbb {Z}[T^{*}] is the group ring of the maximal torus T in Section REF and \lambda _{i}\in T^{*}. Since (T^{*})^{W}=0, the restriction of c_{2} induce... | {
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dcca2709f35994b36ee7274468c7b39e2850ba69 | subsection | 13 | 89 | The group | In the present section, we shall compute the group Q(G) for types B, C, and D. | {
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b8468e0b8e21cba36e94622f8b1f89cb755fe760 | subsection | 14 | 89 | Type | Let G=(\prod _{i=1}^{m}\operatorname{\mathbf {Spin}}_{2n_{i}+1})/{\mu } be an (arbitrary) split semisimple group of type B, m, n_{i}\ge 1, where {\mu }\simeq ({\mu }_{2})^{k} is a central subgroup for some k\ge 0. Let T be the split maximal torus of G (i.e., T=(\operatorname{\mathbb {G}}_m^{\sum n_{i}})/{\mu }) and let... | {
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} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
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247e392fde68bdd3043ff664fd3fbaf37c96ed0a | subsection | 15 | 89 | Type | Then, it follows from (REF ) thatT^{*}=\lbrace \sum a_{i,j}w_{i,j}\,|\, f_{p}(a_{1},\ldots , a_{m})\equiv 0 \mod {2} \rbrace .Let I=\lbrace 1,\ldots , m\rbrace and let I_{1}=\lbrace i\in I\,|\, f_{p}(e_{i})=0,\, 1\le p\le k \rbrace , where \lbrace e_{1}, \ldots , e_{m}\rbrace denotes the standard basis of \mathbb {Z}^{... | {
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f5f33f066d9c411dbbd330a98cf77f18478ac916 | subsection | 16 | 89 | Type | Then, we have\sum a_{i,j}w_{i,j}=\!\sum _{1\le i\le m, 1\le j\le n_{i}-1} a_{i,j}w_{i,j}+\sum _{i\in I_{1}}a_{i}w_{i}+\sum _{p=1}^{k} 2c_{p}w_{i_{p}}+\sum _{s=1}^{l} a_{j_{s}}(w_{j_{s}}+g_{s})where g_{s}=(w_{i_{1}},\ldots , w_{i_{k}})\cdot B_{s} and B_{s} is the s-th column of B, thus we obtain the following \mathbb {Z... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.00886632688343525,
0.03836479410529137,
-0.03689979016780853,
-0.03915833681821823,
0.03894469141960144,
-0.005505210720002651,
0.0012513576075434685,
0.020479535683989525,
-0.027041533961892128,
0.027270440012216568,
-0.005638739559799433,
0.003822745056822896,
-0.022951729595661163,
0... | |
da77627808b673945cd1b0e501d71fa39c7a5b05 | subsection | 17 | 89 | Type | Therefore, with respect to the basis (REF ) we haveq=q^{\prime }+\tfrac{1}{4}\sum _{p=1}^{k} v_{p}^{2}[\delta _{i_{p}}d_{i_{p}}+h_{p}(\delta _{j_{1}}d_{j_{1}}, \ldots , \delta _{j_{l}}d_{j_{l}})]+\tfrac{1}{2}\sum _{1\le i<j\le k} v_{i}v_{j}h_{i}(\delta _{j_{1}}d_{j_{1}}b_{j_{1}}, \ldots , \delta _{j_{l}}d_{j_{l}}b_{j_{... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.03810976445674896,
0.03115297667682171,
-0.015362902544438839,
-0.055654291063547134,
-0.012952437624335289,
0.005110797006636858,
0.03133605048060417,
0.02094663865864277,
-0.017346197739243507,
0.017041075974702835,
-0.030573245137929916,
0.004611159209161997,
-0.019680382683873177,
0... | |
b20702f72be2b6a04da7896a875a0b97150b1525 | subsection | 18 | 89 | Type | We first choose \lbrace w_{i}\rbrace _{i\in I_{1}} as a part of basis of T^{*}. Then, for the remaining part of a basis of T^{*} we write a given basis of R as(e_{j_{1}},\ldots , e_{j_{l}})^{T}=C(e_{i_{1}},\ldots , e_{i_{k}})^{T}for some i_{1},\ldots , i_{k}, j_{1},\ldots , j_{l} with \lbrace i_{1},\ldots , i_{k}, j_{1... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.014894778840243816,
0.03326907753944397,
-0.039617668837308884,
-0.004738554358482361,
0.03049156628549099,
-0.03250602260231972,
0.03595501929521561,
0.00036125752376392484,
0.005635140463709831,
0.039770279079675674,
-0.04437911510467529,
-0.027744578197598457,
-0.03357429802417755,
0... | |
dad63472567d589e15405f90f3a01fcccfbb2d90 | subsection | 19 | 89 | Type | Let T be the split maximal torus of G and let R be the subgroup of (\mathbb {Z}/2\mathbb {Z})^{m} as in (REF ). Then, we have the same commutative diagram (REF ), replacing the middle vertical map (REF ) by\sum a_{i,j}e_{i,j}\mapsto (\sum _{j=1}^{n_{1}} \bar{a}_{1,j}, \ldots , \sum _{j=1}^{n_{m}} \bar{a}_{m,j}),where e... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.004665083717554808,
0.0007476340397261083,
-0.037839438766241074,
0.008056138642132282,
0.021894996985793114,
-0.008658822625875473,
-0.03378085419535637,
0.02796761691570282,
-0.007464897353202105,
0.03851078078150749,
-0.03137011453509331,
0.010985643602907658,
-0.01599021442234516,
0... | |
94de1ea885b13f933a7a84c10e8c00a006b08191 | subsection | 20 | 89 | Type | Since the normalized Killing forms are given byq_{i}=e_{i,1}^{2}+\cdots +e_{i,n_{i}}^{2},for any q\in Q(G) there exist d_{i}\in \mathbb {Z} such that q=\sum _{i=1}^{m}d_{i}q_{i}, thus with respect to the basis (REF ) we haveq=q^{\prime }+\tfrac{1}{4}\sum _{p=1}^{k} v_{p}^{2}[n_{i_{p}}d_{i_{p}}+h_{p}(n_{j_{1}}d_{j_{1}},... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.03735877946019173,
0.008759778924286366,
-0.03839652240276337,
-0.049933794885873795,
0.009629652835428715,
-0.020663311704993248,
0.037236690521240234,
0.01735168881714344,
-0.024676939472556114,
0.033635109663009644,
-0.030674487352371216,
-0.007405370939522982,
-0.015543267130851746,
... | |
800c7c31402809cc81a957ad4b26eb0da5fa2980 | subsection | 21 | 89 | Type | Then, we haveQ(G)=\lbrace \sum _{i=1}^{m}d_{i}q_{i}\,|\,f_{p}(\delta _{1}n_{1}d_{1},\ldots , \delta _{m}n_{m}d_{m})\equiv 0 \mod {4}\rbrace .Let G=(\prod _{i=1}^{m}\operatorname{\mathbf {Spin}}_{2n_{i}})/{\mu } be a split semisimple group of type D, m\ge 1, n_{i}\ge 3, where {\mu }\simeq ({\mu }_{2})^{k_{1}}\times ({\m... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.043793268501758575,
0.010368475690484047,
-0.020813247188925743,
-0.04061939939856529,
0.023895559832453728,
0.0036583400797098875,
0.0051956819370388985,
-0.014755431562662125,
-0.013878040947020054,
0.05624459311366081,
-0.030029669404029846,
-0.008155924268066883,
-0.02554352954030037,... | |
1e5f2fc2f642d24fe3d4fd135ae66886303ca47d | subsection | 22 | 89 | Type | Then, we have the same diagram (REF ), replacing the middle vertical map (REF ) by \prod _{i=1}^{m}\mathbb {Z}^{n_{i}}\rightarrow Z,\sum _{j=1}^{n_{i}} a_{i,j}w_{i,j}\mapsto A_{i}:={\left\lbrace \begin{array}{ll} \big (\overline{a_{i,n_{i}-1}-a_{i, n_{i}}+2S_{i}}\big )e_{i} & \text{ if } n_{i} \text{ odd},\\ \big (\ove... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.023630674928426743,
-0.008306587114930153,
-0.008535418659448624,
-0.04146428406238556,
0.011883987113833427,
-0.010083844885230064,
0.0036327014677226543,
0.045400187373161316,
-0.015461388044059277,
0.021586446091532707,
-0.027734387665987015,
0.0036536776460707188,
-0.04573580622673035... | |
e52fa3b1ad38b50d8f2e454f009b74310200b254 | subsection | 23 | 89 | Type | In view of the argument in the case of type B we may assume that each relation f_{p}(\sum _{i=1}^{m} A_{i})=0 can be written as\delta _{p}a_{p}=b_{p}+4c_{p}, \text{ where } b_{p}={\left\lbrace \begin{array}{ll} \delta _{p}a_{p}+f_{p}(\sum _{i=1}^{m} A_{i}) & \text{ if }a_{p}=a_{i, n_{i}} \text{ with odd } n_{i}, \\ \de... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.016445593908429146,
0.009733106940984726,
0.0005053435452282429,
0.004492789972573519,
-0.02544642984867096,
0.028009381145238876,
0.040824130177497864,
0.0565679632127285,
0.03804760053753853,
-0.013409719802439213,
-0.03478289023041725,
0.004290652461349964,
-0.0014588219346478581,
-0... | |
4798c7713606c424df8360b7806541e29f278de5 | subsection | 24 | 89 | Type | Then, we obtain the following \mathbb {Z}-basis of T^{*}:\lbrace w_{i,j}\rbrace _{i\in I_{1}, \forall j}\cup \lbrace w_{i, 2j}\rbrace _{i\in I^{\prime }, 1\le j\le [\tfrac{n_{i}-2}{2}]}\cup \lbrace \tfrac{4\, }{\delta _{p}}w_{p}\rbrace _{1\le p\le k}\cup \lbrace w_{i,l}+g_{i,l}\rbrace _{w_{i,l}\in W^{\prime }}.Let v_{p... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.019876183941960335,
0.035389676690101624,
-0.02246939204633236,
-0.03169816732406616,
0.003908880986273289,
-0.024925313889980316,
0.037555765360593796,
0.022927017882466316,
-0.01028131041675806,
0.008344030939042568,
-0.013637227937579155,
0.00880165584385395,
-0.04017948359251022,
0.... | |
b8024186a06d01c3e6f0b672283d59f17d6cb9c5 | subsection | 25 | 89 | Type | Hence, q=\sum _{i=1}^{m}d_{i}q_{i}\in Q(G) if and only if&\delta _{p}^{2}[d_{i_{p}}+\sum _{w_{i,l}\in W^{\prime }}d_{i}s_{p}(i,l)^{2}]\equiv 0 \mod {1}6, \sum _{w_{i,l}\in W^{\prime }}d_{i}\delta _{p}\delta _{u}s_{p}(i,l)s_{u}(i,l)\equiv 0 \mod {8}, \\ &\text{ and } d_{i}\delta _{p}s_{p}(i,l)\equiv 0 \mod {2}for all 1\... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.04507746547460556,
0.013947462663054466,
-0.0164042916148901,
-0.05667492002248764,
-0.03528067097067833,
0.0235763993114233,
-0.0091177336871624,
0.02688777819275856,
-0.020951714366674423,
-0.0014487277949228883,
-0.01511483732610941,
-0.04251381754875183,
-0.004059108439832926,
-0.02... | |
1b383a0481f53acfbd5cd418cc2de9c41949d169 | subsection | 26 | 89 | Type | Since\delta _{p}^{2}+\sum _{l}\delta _{p}^{2}s_{p}(i_{p},l)^{2}=\sum _{l}\delta _{p}^{2}s_{p}(i,l)^{2}={\left\lbrace \begin{array}{ll} 8 & \text{ if } c_{i}(p)=2 \text{ or } c_{i,1}(p)+c_{i,2}(p)=4,\\
2n_{i} & \text{ if } c_{i}(p)=\pm 1 \text{ or } c_{i,1}(p)+c_{i,2}(p)=2\end{array}\right.}for all p and i\ne i_{p}, whe... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.05315437912940979,
0.009115884080529213,
-0.029964178800582886,
0.0015495095867663622,
-0.04229160025715828,
0.056815989315509796,
0.01795714721083641,
0.07402555644512177,
0.02364790067076683,
-0.0014064779970794916,
-0.019284481182694435,
-0.011518816463649273,
-0.019635386765003204,
... | |
df98d0bf014018fa8acd9515814f9bf13dac52d9 | subsection | 27 | 89 | Type | Since we have\sum _{l}s_{p}(i,l)s_{u}(i,l)\equiv {\left\lbrace \begin{array}{ll} \pm 2n_{i} \mod {8} & \text{ if } c_{i}(p)c_{i}(u)\equiv \pm 1 \mod {4},\\ 4\,\,\,\,\,\,\,\,\, \mod {8} & \text{ if } c_{i}(p)c_{i}(u)\equiv 2 \mod {4},\\ 0\,\,\,\,\,\,\,\,\, \mod {8} & \text{ otherwise}\end{array}\right.}for all 1\le p<u\... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.04235943406820297,
0.016556190326809883,
-0.017319148406386375,
-0.04583852365612984,
-0.0001521147642051801,
0.02659671939909458,
0.02273615077137947,
0.020569350570440292,
-0.012649845331907272,
0.041199736297130585,
-0.02264459617435932,
-0.00785083882510662,
-0.016189970076084137,
0... | |
7bd654d3b95af44fa68dee901264375e93f2d505 | subsection | 28 | 89 | Type | As the Weyl group of \operatorname{\mathbf {Spin}}_{2n+1} contains a normal subgroup (\mathbb {Z}/2\mathbb {Z})^{n} generated by sign switching, we see that 2 \,|\,c_{2}(\rho (\lambda )) for any \lambda \in \Lambda (c.f. ), thus \operatorname{Dec}(\operatorname{\mathbf {Spin}}_{2n+1})=2\mathbb {Z}q. Therefore,\operator... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1090/ulect/028",
"end": 301,
"openalex_id": "https://openalex.org/W1534617670",
"raw": "S. Garibaldi, A. Merkurjev, J.-P. Serre, Cohomological Invariants in Galois Cohomology, University Lecture Series 28, AMS, Providence, RI, (2003).",
... | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.05505909025669098,
0.02952864207327366,
-0.013574019074440002,
-0.018525980412960052,
0.013123841024935246,
-0.008721250109374523,
-0.0136503204703331,
-0.0030062750447541475,
-0.01327644381672144,
0.016923651099205017,
-0.018251294270157814,
0.006077406462281942,
-0.024920037016272545,
... | |
d15e9812a0f55381c956a7ce142d36b41344ec05 | subsection | 29 | 89 | Type | Let R_{1}^{\prime }=\langle e_{i}\in R\,|\, n_{i}\le 2\rangle and R_{2}^{\prime }=\langle e_{i}+e_{j}\in R\,|\, e_{i}, e_{j}\notin R,\, n_{i}=n_{j}=1\rangle be two subspaces of R with \dim R_{1}^{\prime }=l_{1} and \dim R_{2}^{\prime }=l_{2}. | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.032921697944402695,
0.04396678879857063,
0.018825475126504898,
0.016354059800505638,
-0.02085447683930397,
0.03978674113750458,
-0.029046760872006416,
0.042685315012931824,
-0.01684224046766758,
-0.0016628660960122943,
-0.04784172400832176,
-0.01777283474802971,
-0.0376814603805542,
-0.... | |
f1ae9dfb65c1eff2980f8107f4ee0776f22ef355 | subsection | 30 | 89 | Type | Then,\operatorname{Dec}(G)=(\bigoplus _{e_{i}\in R_{1}^{\prime }}\mathbb {Z}q_{i})\oplus (\bigoplus _{n_{i}\ge 2,\, e_{i}\notin R_{1}^{\prime }} 2\mathbb {Z}q_{i})\oplus (\bigoplus _{r=1}^{l_{2}} 2\mathbb {Z}q^{\prime }_{r})\oplus (\bigoplus _{s=1}^{l_{3}} 4\mathbb {Z}q^{\prime \prime }_{s}),where l_{3}=m-l_{1}-l_{2}-|... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.024518638849258423,
-0.004783194046467543,
-0.012671268545091152,
-0.03286443278193474,
-0.006896343547850847,
0.015059051103889942,
-0.004577219486236572,
0.014708131551742554,
-0.011649022810161114,
-0.013289193622767925,
-0.03081994317471981,
0.008750117383897305,
-0.017378175631165504... | |
a08b1efd857a021a2f64e4beb25c7b4a79130685 | subsection | 31 | 89 | Type | Let us denote the right hand side of equation (REF ) by D. We write D=\bigoplus D_{u}, where D_{u} denotes u-th direct summand of D for 1\le u\le 4. First, we show that D\subseteq \operatorname{Dec}(G). If e_{i}\in R_{1}^{\prime }, then by (REF ) we have w_{i,1}, w_{i,2}\in T^{*}, thus by (REF ) and (REF ) D_{1}\subset... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.0186738483607769,
0.022930387407541275,
-0.024654362350702286,
-0.034448977559804916,
0.009619168005883694,
-0.017987309023737907,
0.02828538790345192,
0.013890963047742844,
0.006369552109390497,
0.03466256707906723,
-0.03338102996349335,
-0.008337629027664661,
-0.01600397564470768,
0.0... | |
ceb76cb2a39c3ae26e0a235a7eb23ceb97550711 | subsection | 32 | 89 | Type | If both a_{i} and a_{j} are even, then \lambda \in (\Lambda _{i})_{r}\oplus (\Lambda _{j})_{r}, so c_{2}(\rho (\lambda ))\in D_{3}\oplus D_{4}. If a_{i} is even and a_{j} is odd, then as \lambda \in T^{*} if and only if e_{j}\in R_{1}^{\prime }, we get c_{2}(\rho (\lambda ))\in D_{1}\oplus D_{3}\oplus D_{4}. Similarly,... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.03201408311724663,
0.03241082653403282,
-0.03259393572807312,
-0.015350889414548874,
0.002592179225757718,
-0.022522777318954468,
-0.009933826513588428,
0.03274653106927872,
-0.0040017603896558285,
0.025971386581659317,
-0.026703834533691406,
-0.013031471520662308,
-0.022202331572771072,
... | |
10cadf2d4039689b0a94e8e1edf3b1e216c6c942 | subsection | 33 | 89 | Type | Moreover, since the Weyl group of \operatorname{\mathbf {Sp}}_{2n} contains a normal subgroup (\mathbb {Z}/2\mathbb {Z})^{n} generated by sign switching, we see that \tfrac{4}{\gcd (2, n)} \,|\,c_{2}(\rho (\lambda )) for any \lambda \in \Lambda _{r} (c.f. ), thus \operatorname{Dec}(\operatorname{\mathbf {PGSp}}_{2n})=\... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1090/ulect/028",
"end": 361,
"openalex_id": "https://openalex.org/W1534617670",
"raw": "S. Garibaldi, A. Merkurjev, J.-P. Serre, Cohomological Invariants in Galois Cohomology, University Lecture Series 28, AMS, Providence, RI, (2003).",
... | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.06483635306358337,
0.02113897167146206,
-0.022283680737018585,
-0.02831248566508293,
0.0136144133284688,
-0.0009219682542607188,
-0.015362003818154335,
-0.004159111995249987,
0.0017618989804759622,
0.016178563237190247,
-0.02794617787003517,
-0.013019165024161339,
-0.016086986288428307,
... | |
c0fe5d0a51f357a41551db83f260c190578a78ce | subsection | 34 | 89 | Type | Then,\operatorname{Dec}(G)= (\bigoplus _{e_{i}\in R}\mathbb {Z}q_{i})\oplus (\bigoplus _{n_{i}\equiv 0\!\!\!\!\mod {2},\, e_{i}\notin R} 2\mathbb {Z}q_{i}) \oplus (\bigoplus _{r=1}^{l_{2}} 2\mathbb {Z}q^{\prime }_{r})\oplus (\bigoplus _{s=1}^{l_{3}} 4\mathbb {Z}q^{\prime \prime }_{s}),where l_{3}=|\lbrace i \,|\, n_{i}... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.0031756702810525894,
0.0011214979458600283,
-0.021697552874684334,
-0.04391389340162277,
-0.0026015702169388533,
0.03640672564506531,
-0.0004155550559516996,
0.002317381091415882,
-0.015700971707701683,
0.014304821379482746,
-0.030455918982625008,
-0.006194941233843565,
-0.015990883111953... | |
91775a1669766f89a629566e11ed00619e72e6e4 | subsection | 35 | 89 | Type | Similarly, by (REF ) we have D_{2}\oplus D_{4}\subseteq \operatorname{Dec}(G). Let e_{i}+e_{j}\in R_{2}^{\prime \prime }. Then, by (REF ) we have e_{i,1}+e_{j,1}\in T^{*}. As both n_{i} and n_{j} are odd, by (REF ) we get 2q_{i}+2q_{j}\in \operatorname{Dec}(G), i.e., D_{2}\subseteq \operatorname{Dec}(G). Therefore, we ... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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... | |
6d9b742b966ef4f0dfe0f4e921803a45bde786ab | subsection | 36 | 89 | Type | Then, as before it follows from the action of the normal subgroups (\mathbb {Z}/2\mathbb {Z})^{n_{i}} of W thatc_{2}(\rho (\lambda ))=4(\sum _{i\in J} a_{i}q_{i})+2(\sum _{i\in K}b_{i}q_{i})for some a_{i}, b_{i}\in \mathbb {Z}. Therefore, we get c_{2}(\rho (\lambda ))\in D, thus \operatorname{Dec}(G)\subseteq D.Let G=(... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1090/ulect/028",
"end": 1635,
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"raw": "S. Garibaldi, A. Merkurjev, J.-P. Serre, Cohomological Invariants in Galois Cohomology, University Lecture Series 28, AMS, Providence, RI, (2003).",... | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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0... | |
fe83ed70c0c411be7eee8f590d4fba0cb33f196f | subsection | 37 | 89 | Type | Hence, by (REF ) we obtain\delta _{1}^{\prime \prime }\mathbb {Z}q_{1}\oplus \cdots \oplus \delta _{m}^{\prime \prime }\mathbb {Z}q_{m}\subseteq \operatorname{Dec}(G)\subseteq \delta _{1}^{\prime }\mathbb {Z}q_{1}\oplus \cdots \oplus \delta _{m}^{\prime }\mathbb {Z}q_{m}, \text{ where }\delta _{i}^{\prime \prime }={\le... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.48550/arxiv.1305.2899",
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"raw": "H. Bermudez and A. Ruozzi, Degree 3 cohomological invariants of groups that are neither simply connected nor adjoint, J. Ramanujan Math. Soc.... | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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0.0... | |
afea343578a1024ed089026feb46be76e3f1f1fb | subsection | 38 | 89 | Type | Set&I_{1}^{\prime }=\lbrace i \,\,|\,\, R_{1,i}\ne 0, n_{i}\ne 3\rbrace \cup \lbrace i \,\,|\,\,R_{1,i}=2Z_{i}, n_{i}=3\rbrace \cup \lbrace i \,\,|\,\, R_{1,i}^{\prime }\ne 0,\, n_{i}=4\rbrace \,\cup \\ &\lbrace i \,\,|\,\, e_{i,1}+e_{i,2}\in R_{1,i}^{\prime },\, n_{i}\ge 6\rbrace ,\,\, I_{2}^{\prime }=\lbrace i \,\,|\... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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1d686479ea6012b6837a1c31c84b62c220dc1408 | subsection | 39 | 89 | Type | We denote by D the right hand side of equation (REF ) and we write D=\bigoplus D_{u}, where D_{u} denotes u-th direct summand of D for 1\le u\le 5. If e_{i}\in R with n_{i}=3, then by (REF ) D_{1}\subseteq \operatorname{Dec}(G). If 2e_{i}\in R or e_{i,1}+e_{i,2}\in R, then by (REF ) we have w_{i,1}\in T^{*}, thus by (R... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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... | |
d03dae31b2d9986fca4136061abf0967a53d8c8e | subsection | 40 | 89 | Type | Applying the same argument as in the proof of Proposition REF we obtainc_{2}(\rho (\lambda ))\in {\left\lbrace \begin{array}{ll} D_{4}\oplus D_{5} & \text{ if } A_{i}=0 \text{ with odd }n_{i},\\ D_{2} & \text{ if } A_{i}\ne 0 \text{ with odd } n_{i}\ge 5 ;\text{ or } A_{i}=2e_{i} \text{ with } n_{i}=3,\\ D_{1} & \text{... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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c7f2adedf964307f2d7e7224aebf0374d8c90083 | subsection | 41 | 89 | Type | Then,\operatorname{Inv}^{3}(G)_{\operatorname{red}}=(\mathbb {Z}/2\mathbb {Z})^{m-k-l_{1}-l_{2}},\,\, \text{where}l_{1}=\dim \langle e_{i}\in R\,|\, n_{i}\le 2\rangle and l_{2}=\dim \langle e_{i}+e_{j}\in R\,|,\, e_{i}, e_{j}\notin R,\, n_{i}=n_{j}=1\rangle .Let R=\lbrace r=(r_{1},\ldots , r_{m})\in (\mathbb {Z}/2\math... | {
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{
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"raw": "D. Laackman and A. Merkurjev, Degree three cohomological invariants of reductive groups, Comment. Math. Helv. 91 (2016), no. 3, 493–518.",
"sour... | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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... | |
fd75355ab8ed7ddece70196302118294c689376a | subsection | 42 | 89 | Type | Therefore, any reductive indecomposable invariant of G corresponding to q=\sum _{i=1}^{m}d_{i}q_{i}\in Q(G) satisfiesf_{p}(\frac{\delta _{1}d_{1}}{2}, \ldots , \frac{\delta _{m}d_{m}}{2})\equiv 0 \mod {2}, \text{ where } \delta _{i}={\left\lbrace \begin{array}{ll}
2 & \text{ if } n_{i}\ge 2 \text{ or } e_{i}\in R,\\
1 ... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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6457981f538c32a2e9496ff7ca5841b68fe3242a | subsection | 43 | 89 | Type | Then,\operatorname{Inv}^{3}(G)_{\operatorname{\hspace{0.85358pt}ind}}=\operatorname{Inv}^{3}(G)_{\operatorname{red}}=(\mathbb {Z}/2\mathbb {Z})^{m-k-l},where l=\dim \langle e_{i}\in R\,|\, n_{i}=2\rangle .By Theorem REF , it suffices to show that \operatorname{Inv}^{3}(G)_{\operatorname{\hspace{0.85358pt}ind}}\subseteq... | {
"cite_spans": [
{
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"doi": "10.1070/im8473",
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"source_ref_id": "1388985b... | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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0.006159564014524221,... | |
2644aad017368f3ee0ea1470ab98f82427f64a5f | subsection | 44 | 89 | Type | Then,\operatorname{Inv}^{3}(G)_{\operatorname{red}}=(\mathbb {Z}/2\mathbb {Z})^{s+l-l_{1}-l_{2}},\,\, \text{where}l_{1}=\dim \langle e_{i}\,|\, e_{i}\in R\rangle , l_{2}=\dim \langle e_{i}+e_{j}\,|\, e_{i}+e_{j}\in R,\, e_{i}, e_{j}\notin R,\, n_{i}\equiv n_{j}\equiv 1\mod {2}\rangle , and l=\dim \big (R\cap (\bigoplus... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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1daa2b81c8fea1d2bea26346956ec93a055faaac | subsection | 45 | 89 | Type | Therefore, any reductive indecomposable invariant of G corresponding to q=\sum _{i=1}d_{i}q_{i}\in Q(G) obviously satisfies the first equation of (REF ) and the second equation of (REF ) divided by 2, i.e.,f_{p}(\tfrac{\delta _{1}n_{1}d_{1}}{2},\ldots , \tfrac{\delta _{m}n_{m}d_{m}}{2})\equiv 0 \mod {2}, \text{ where }... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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... | |
87f103ae1c56a3a933a59148899fa3d5373b9fb4 | subsection | 46 | 89 | Type | Let R be the subgroup of the character group Z defined in ( REF) such that {\mu }^{*}=Z/R, R_{1,i}^{}=R\cap Z_{i} for odd n_{i}, R^{\prime }_{1,i}=R\cap Z_{i} for even n_{i}, and let\bar{R}=\lbrace (\bar{r}_{1},\ldots , \bar{r}_{m})\in \bigoplus _{i=1}^{m}(\mathbb {Z}/2\mathbb {Z})\bar{e}_{i}\,|\, \sum _{i=1}^{m}r_{i}\... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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0... | |
90c5586b452da13e00cb9c7ca28c4466d1251f92 | subsection | 47 | 89 | Type | Set&R^{\prime }=\bar{R}\cap \big (\bigoplus _{4\nmid n_{i}, R_{1,i}^{\prime }, R_{1,i}\ne Z_{i}}(\mathbb {Z}/2\mathbb {Z})\bar{e}_{i}\big ) \text{ with } l=\dim R^{\prime }, \, I_{1}=\lbrace i\,|\, Z_{i}=R_{1,i} \text{ or } R_{1,i}^{\prime }, n_{i}\ne 3\rbrace ,\\
&I_{2}=\lbrace i \,|\,R_{1,i}^{\prime }=0, 4|n_{i}\rbra... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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13c7e84a0f61b2c13066dd5c4ec6a824d5054738 | subsection | 48 | 89 | Type | Then, \theta _{i,j}=1 for all 1\le i\le m and 1\le j\le n_{i}. Note that the order of the fundamental weight w_{i,j} in \Lambda /T^{*} is trivial for all j if and only ifZ_{i}={\left\lbrace \begin{array}{ll} R_{1,i} & \text{ if } n_{i} \text{ odd,}\\ R_{1,i}^{\prime } & \text{ if } n_{i} \text{ even.}\end{array}\right.... | {
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"sourc... | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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0a33467d825900036b96aa633b7dadda93f0df22 | subsection | 49 | 89 | Type | Therefore, it follows by (REF ) that\operatorname{Inv}^{3}(G)_{\operatorname{red}}=\frac{\lbrace \sum _{i=1}^{m}d_{i}q_{i}\,|\, \bar{f}_{p}(\epsilon _{1} d_{1},\cdots , \epsilon _{m}d_{m})\equiv 0 \mod {2} \rbrace }{\operatorname{Dec}(G)}where, \bar{f}_{p}\in \mathbb {Z}/2\mathbb {Z}[t_{1},\ldots , t_{m}] denotes the i... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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fab8b32b9b1abeba915a8d247d6d21c26b01b945 | subsection | 50 | 89 | Type | Observe that \bar{f}_{p}(\bar{e}_{i})\equiv 0 \mod {2} for all p with n_{i} odd if and only if either c_{i}(p)=0 or 2 for all p (i.e., f_{p}(e_{i})\equiv 0 or f_{p}(2e_{i})\equiv 0 \mod {4}, respectively) and this, in turn, is equivalent to R_{1,i}=Z_{i} or 2Z_{i}. Similarly, \bar{f}_{p}(\bar{e}_{i})\equiv 0 \mod {2} f... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
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06777e6aec5c61d1b0b4de7dee5d3f53aa1c4b66 | subsection | 51 | 89 | Type | Therefore, the statement immediately follows by Proposition REF .Lemma 6.1
Let G=(\prod _{i=1}^{m}\operatorname{\mathbf {Spin}}_{2n_{i}+1})/{\mu }, m, n_{i}\ge 1, where {\mu } is a central subgroup. Let R be the subgroup of ({\mu }_{2}^{m})^{*}=(\mathbb {Z}/2\mathbb {Z})^{m} whose quotient is the character group {\mu ... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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0... | |
18786f49829606ab5c13b036f5f2f8a331ae1cc4 | subsection | 52 | 89 | Type | Consider the natural exact sequence1\rightarrow (\operatorname{\mathbb {G}}_m)^{m}/{\mu }\rightarrow G_{\operatorname{red}}\rightarrow \prod _{i=1}^{m}\operatorname{\mathbf {O}}^{+}_{2n_{i}+1}\rightarrow 1.Then, by Hilbert Theorem 90 and , this sequence yields a bijection between the set H^{1}(F,G_{\operatorname{red}})... | {
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"Sanghoon Baek"
] | [
"math.AG"
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1975c6d921dc76e9c61b655fbf8f555abafa23d0 | subsection | 53 | 89 | Type | Since (C_{0}(\phi _{1}), \ldots , C_{0}(\phi _{m}))\in \operatorname{Ker}(\tau ) if and only if it is contained in the kernel of the compositionH^{2}(F, ({\mu }_{2})^{m})\stackrel{\tau }{\rightarrow }H^{2}(F,({\mu }_{2})^{m}/{\mu })\stackrel{r_{*}}{\rightarrow }H^{2}(F, \operatorname{\mathbb {G}}_m)for all r\in R=(({\m... | {
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"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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... | |
13c8fec9bfebf19c4bc25be239909f437d945117 | subsection | 54 | 89 | Type | Since C_{0}(\phi \perp \langle 1 \rangle )=C_{0}(\phi ) for any odd-dimensional quadratic form \phi and \operatorname{disc}(-\phi _{i_{s}}\perp \langle 1\rangle )=1, the same argument shows that
(-\phi _{i_{1}}\perp \phi _{i_{2}})\perp \cdots \perp (-\phi _{i_{s-2}}\perp \phi _{i_{s-1}})\perp (-\phi _{i_{s}}\perp \lang... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
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0.0... | |
8a3985a1fe51900ed5e7dfcad6885f5907f69eab | subsection | 55 | 89 | Type | Moreover, we have\operatorname{Inv}^{3}(G_{\operatorname{red}})_{\operatorname{norm}}\simeq \frac{R}{\langle e_{i},\, e_{j}+e_{k}\in R\,|\, e_{j}, e_{k}\notin R,\, n_{i}\le 2,\, n_{j}=n_{k}=1\rangle }.Observe that \operatorname{Inv}^{3}(G_{\operatorname{red}})_{\operatorname{norm}}=\operatorname{Inv}^{3}(G_{\operatorna... | {
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"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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fe2313621c2d25c990690549922c2a7f2857309e | subsection | 56 | 89 | Type | Otherwise, by the proof of Lemma REF each invariant {\mathrm {e}}_{3}(\phi [r]) is nontrivial, thus the statements follow from Theorem REF .Recall from Section the following subgroups of R.R_{1}=\langle e_{i}\in R\rangle \text{ and } R_{2}=\langle e_{i}+e_{j}\in R\,|\, e_{i}, e_{j}\notin R_{1}\rangle .We shall need the... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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... | |
7e01a770ad5a3492db2cef62b88c83387d4dabf1 | subsection | 57 | 89 | Type | Then, we have
\partial _{z}(\alpha (\phi ))=(x, y)\ne 0, where \partial _{z} denotes the residue map, thus \alpha (\phi ) ramifies.Now we may assume that \alpha (\phi )={\mathrm {e}}_{3}(\phi [r_{2}])+{\mathrm {e}}_{3}(\phi [r_{3}]) with r_{2}\ne 0. To show that \alpha (\phi ) ramifies, we shall choose bases of R_{2} a... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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0.... | |
7e3dc275fb319e90d4b0c986b4b3bf2995276e10 | subsection | 58 | 89 | Type | We can divide all elements of the basis C_{3} into two types: either e(i_{p}) for some i_{p}\in J_{2} appears in e(k_{1},\ldots , k_{l})\in C_{3} (the first type) or not (the second type).We first select basis elements from the first type elements as follows. We choose any element b(i_{1}) in C_{3} of the first type su... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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a6cef133073af649d706bfba830944ae2c60892b | subsection | 59 | 89 | Type | Now we choose another element b(j_{2}) of the second type for some j_{2}\notin J_{2}, so that we have b(j_{2}):=e(j_{2})+b^{\prime }(j_{2}), where both e(j_{1}) and e(j_{2}) do not appear in b^{\prime }(j_{2}). Again we modify every element of C_{3} by adding b(j_{2}) to the element whenever e(j_{2}) appears in the ele... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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... | |
cc01a32042e3295b640e738c62b41772302d182d | subsection | 60 | 89 | Type | Let e(i_{u}, i_{u,v})\in B_{2}^{\prime } be such an element for some 1\le u\le k and 1\le v\le m_{u} and let I=\lbrace 1,\ldots , m\rbrace . We take a division quaternion algebra (x, y) over a field extension K/F. Then, choose \phi _{i} for all i\in I such that\phi [e(i_{u})]=\phi [e(i_{u,q})]=\langle x, y, xy\rangle \... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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c3a74089473e2d35a653284d7e6bbef32ef3ee0d | subsection | 61 | 89 | Type | Observe that by construction of B_{3} there existsk_{1}\in I\backslash \lbrace i_{p}, j_{r}\,|\, i_{p}\in J_{2}^{\prime }, 1\le r\le s\rbracesuch that e(k_{1}) appears in b^{\prime }(i_{u}). We first choose \phi [e(i_{u,v})] as in (REF ) and \phi [e(k_{1})]=\langle x, y, xy\rangle \perp h. Then, we choose \phi _{i} for... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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... | |
38014fc5f5df4c0db7500da868b3ec8e925cf7ea | subsection | 62 | 89 | Type | Hence, \partial _{z}(\alpha (\phi ))=(x, y)\ne 0, thus \alpha (\phi ) ramifies.We present the second main result on the group of unramified degree 3 invariants for type B.Theorem 6.5
Let G=(\prod _{i=1}^{m}\operatorname{\mathbf {Spin}}_{2n_{i}+1})/{\mu } defined over an algebraically closed field F, m, n_{i}\ge 1, whe... | {
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"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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972184bdbbc9ef57857154dccda6589a2bd934c7 | subsection | 63 | 89 | Type | Let R be the subgroup of ({\mu }_{2}^{m})^{*}=(\mathbb {Z}/2\mathbb {Z})^{m} whose quotient is the character group {\mu }^{*}. Set G_{\operatorname{red}}=(\prod _{i=1}^{m}\operatorname{\mathbf {GSp}}_{2n_{i}})/{\mu }, where \operatorname{\mathbf {GSp}}_{2n_{i}} denotes the group of symplectic similitudes. Then, for any... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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... | |
8797ae192e6e6ed9a5f3f4a118e9d649a089b984 | subsection | 64 | 89 | Type | Since (A_{1}, \ldots , A_{m})\in \operatorname{Ker}(\tau ) if and only if it is contained in the kernel of the map in (REF ) for all r\in R, thus we haveH^{1}(F, G_{\operatorname{red}})\simeq \lbrace \big ((A_{1},\sigma _{1}),\ldots , (A_{m},\sigma _{m})\big )\,|\,\deg A_{i}=2n_{i}, \sum _{i=1}^{m}r_{i}A_{i}=0 \rbracef... | {
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"source... | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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0cf318c34066761a516327caa9c15a628428af60 | subsection | 65 | 89 | Type | Set G_{\operatorname{red}}=(\prod _{i=1}^{m}\operatorname{\mathbf {GSp}}_{2n_{i}})/{\mu }. Then, every normalized invariant in \operatorname{Inv}^{3}(G_{\operatorname{red}}) is of the form\sum _{r\in R^{\prime }}{\mathrm {e}}_{3}(\phi [r])+\sum _{i\in I^{\prime }}\Delta _{i}for some R^{\prime }\subseteq R\cap (\bigoplu... | {
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"source... | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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2724f2fb28cb8a893d548f542068c003a8698d08 | subsection | 66 | 89 | Type | As e_{i}\notin R, it follows by the rank theorem (or Rouché-Capelli theorem) that there exists a G_{\operatorname{red}}-torsor \eta =\big ((A_{1},\sigma _{1}),\ldots , (A_{m},\sigma _{m})\big ) over E such that(A_{i},\sigma _{i})=(M_{n_{i}}(Q), \sigma _{b}\otimes \gamma ) \text{ and } (A_{j},\sigma _{j})=(M_{2n_{j}}(E)... | {
"cite_spans": [
{
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"source_... | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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66a6d0b8f745cc47b7fddfabfaaed3834fa3062a | subsection | 67 | 89 | Type | For any e_{i}\in R_{1}^{\prime \prime } and any e_{j}+e_{k}\in R_{2}^{\prime \prime }, we have\phi _{i}=T_{\sigma _{i}}=h\, \text{ and }\, \phi _{j}\perp \phi _{k}=T_{\sigma _{j}}\perp T_{\sigma _{k}}=\langle \langle a, b, 1\rangle \rangle \perp h^{\prime },where A_{j}=A_{k}=(a, b) in \operatorname{Br}(K), h and h^{\pr... | {
"cite_spans": [
{
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"source_r... | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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f846ee11b830dcf2b372a7df657f0a2f343bc6d8 | subsection | 68 | 89 | Type | Then, every normalized invariant in \operatorname{Inv}^{3}(G_{\operatorname{red}}) is ramified if either n_{i} is divisible by 4 for some i with e_{i}\notin R_{1} or n_{j}n_{k}\lnot \equiv 1 \mod {2} for some j and k such that e_{j}+e_{k}\in R\cap (\bigoplus _{4\nmid n_{i}}(\mathbb {Z}/2\mathbb {Z}) e_{i}).Let \alpha b... | {
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"source_... | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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ccc128f3746d7159d6551bf93c25b3a2a471aa3b | subsection | 69 | 89 | Type | Therefore, by (REF ) we have \partial _{z}(\alpha (\eta ))=(x, y)\ne 0, thus the invariant \alpha ramifies.We may assume that n_{i}\lnot \equiv 0 \mod {4} for all 1\le i\le m, thus\alpha (\eta )={\mathrm {e}}_{3}(\phi [r_{2}])+{\mathrm {e}}_{3}(\phi [r_{3}])for some nonzero r_{2}\in R_{2} and some r_{3}\in R_{3}, where... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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70261597e78c4abc022d81b6229976e23c164858 | subsection | 70 | 89 | Type | Now we choose \eta =\big ((A_{i},\sigma _{i})\big ) for i\in I such that(A_{i}, \sigma _{i})=(M_{d}(Q), t\otimes \gamma ),\,\, (A_{i_{u,v}}, \sigma _{i_{u,v}})=(M_{d}(Q_{1}\otimes Q_{2}), t\otimes \gamma _{1}^{\prime }\otimes \gamma _{2})for i=i_{u}, i_{u,q} and all 1\le q\ne v\le m_{u}, where t denotes the transpose i... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
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d9c72087be3aaee47664ff5ef5052676f299d6ae | subsection | 71 | 89 | Type | We choose k_{1} as in (REF ) and then choose (A_{k_{1}}, \sigma _{k_{1}}) and (A_{i_{u,v}}, \sigma _{i_{u,v}}) as in (REF ). Then, we choose (A_{i}, \sigma _{i}) for i\in I\backslash \lbrace i_{u,v}, k_{1} \rbrace such that(A_{i}, \sigma _{i})={\left\lbrace \begin{array}{ll}(M_{d}(Q), t\otimes \gamma ) & \text{ if } e(... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
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bd0823dfcf1aa9284493cf3033eb556fa0a70cb7 | subsection | 72 | 89 | Type | Then, every unramified degree 3 invariant of G is trivial, i.e., \operatorname{Inv}^{3}_{\operatorname{nr}}(G)=0.Let G_{\operatorname{red}}=(\prod _{i=1}^{m}\operatorname{\mathbf {GSp}}_{2n_{i}})/{\mu }, G^{\prime }_{\operatorname{red}}=(\operatorname{\mathbf {GSp}}_{2})^{m}/{\mu }, and G^{\prime }=(\operatorname{\math... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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b72b871c693bec38622df60696efecfa9027de38 | subsection | 73 | 89 | Type | Then, for any field extension K/F the first Galois cohomology set H^{1}(K, G_{\operatorname{red}}) is bijective to the set of m-tuples \big ((A_{1},\sigma _{1}, f_{1}),\ldots , (A_{m},\sigma _{m}, f_{m})\big ) of triples consisting of a central simple K-algebra A_{i} of degree 2n_{i} with orthogonal involution \sigma _... | {
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{
"arxiv_id": "",
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"raw": "M.-A. Knus, A. Merkurjev, M. Rost, and J.-P. Tignol, The book of involutions, American Mathematical Society, Providence, RI, 1998, With a preface in ... | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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5193e6630d36e9dbfe07b49b1e11e9f865d1cf74 | subsection | 74 | 89 | Type | Consider the exact sequence1\rightarrow (\operatorname{\mathbb {G}}_m)^{2m}/{\mu }\rightarrow G_{\operatorname{red}}\rightarrow \prod _{i=1}^{m}\operatorname{\mathbf {PGO}}^{+}_{2n_{i}}\rightarrow 1,where \operatorname{\mathbf {PGO}}^{+}_{2n_{i}} denotes the projective orthogonal group. Applying the same argument as in... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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e3d9750ba4a26f2a6b64543626aae4f67e41a567 | subsection | 75 | 89 | Type | (C_{i,2}, C_{i,1}))) and the map \tau is induced by the natural surjection Z(\prod _{i=1}^{m}\operatorname{\mathbf {Spin}}_{2n_{i}})\rightarrow Z(\prod _{i=1}^{m}\operatorname{\mathbf {Spin}}_{2n_{i}})/{\mu }. As (B^{\prime }_{1}, \ldots , B^{\prime }_{m})\in \operatorname{Ker}(\tau ) if and only if it is contained in ... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
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9ab9553a111a06c8945d000ffc89e9c54c2518d1 | subsection | 76 | 89 | Type | Hence, the Arason invariant {\mathrm {e}_{3}} induces the following invariant{\mathrm {e}}_{3, i}: H^{1}(K, G_{\operatorname{red}})\rightarrow H^{3}(K)given by {\mathrm {e}}_{3, i}\big ((A_{1},\sigma _{1}, f_{1}),\ldots , (A_{m},\sigma _{m}, f_{m})\big )={\mathrm {e}_{3}}(\psi _{i}). This invariant is obviously nontriv... | {
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"source_ref_id": "671bdf132d... | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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afabc966b5fe03cf414d5409901603dd32b617f8 | subsection | 77 | 89 | Type | Since \sigma _{1}=\operatorname{Int}(x)\circ t for some t-symmetric invertible element x, where \operatorname{Int}(x) denotes the inner automorphism induced by x, we have\operatorname{disc}(\sigma _{1})=\operatorname{Nrd}_{M_{n}(F)}(x)=\sqrt{\operatorname{Nrd}_{A}(x\otimes 1)}and \sigma =\operatorname{Int}(x\otimes 1)\... | {
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{
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"source_ref_id": "671bdf132d5... | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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a34226d5e7c669e0eeb02959a9b4e63ee3565cb0 | subsection | 78 | 89 | Type | Moreover, we have\operatorname{Inv}^{3}(G_{\operatorname{red}})_{\operatorname{norm}}\simeq \frac{\bigoplus _{i\in I_{1}\cup I_{2}}(\mathbb {Z}/2\mathbb {Z}) \bar{e}_{i}\bigoplus R^{\prime }}{\langle \bar{e}_{i},\, \bar{e}_{j}+\bar{e}_{k}\in R^{\prime }\,|\, \bar{e}_{j}, \bar{e}_{k}\notin R^{\prime },\, n_{j}\equiv n_{... | {
"cite_spans": []
} | 1801.08845 | Degree three invariants for semisimple groups of types $B$, $C$, and $D$ | [
"Sanghoon Baek"
] | [
"math.AG"
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