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961b711799cf2d7eeef221806c1d7083f1fdf836 | subsection | 18 | 35 | Simulation M5 | Simulator design and development explores a high-level architecture design space. Simulation enables the user to evaluate various deployment topologies are varying level of abstraction. It examines the architectural building blocks in the context of performance optimization. We use the M5 architectural simulator develo... | {
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86d85937fce2b5ef54b6cb963ca66c4d6e3908e9 | subsection | 19 | 35 | Types of process wait state used to achieve the profiling | Unix POSIX supports a variety of multi-threading and multi-process synchronization primitives. Pthread library is used to implement multi-threading. It supports the creation of threads using pthread_create and synchronization wait using pthread_join. Pthread supports the use of mutually exclusive locks through the use ... | {
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723432b4548f1c537c04a2f4c3225d8d0bef1c38 | subsection | 20 | 35 | Xenoprof architecture | Xenoprof is an open source profiling utility base on OProfile REF . It is developed on the Xen virtual machine environment and enables the user to gather system wide data. Xen is an open source virtual machine monitor (VMM). OProfile can be used to profile kernel and user level applications and libraries. It enables p... | {
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5fcae4201a219cdb4fd201ba1ce40929ed6be56e | subsection | 21 | 35 | Linux perf | Performance data metrics will be gathered using the perf utility. There are a large number of Unix programs that are available for performance profiling including top, iostat and sar. We run the perf utility in user mode to collect data for a specific application. The call-graph option is used to obtain function sub-tr... | {
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89d1b20ce5bb800bf7b3d383f7708bc31639707d | subsection | 22 | 35 | Network drivers | We evaluate the performance of the Linux network drivers using perf events sock, net and skb. scsi information is obtained using the scsi event.>> perf record -e sock:* -e net:* -e skb:* -e scsi:* -e cpu-clock <<application>> | {
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3054f174014b8624bc2dcffd471d017377b4731c | subsection | 23 | 35 | gprof | gprof is used to profile data using compiler annotated binary format. Application is compiled using the –pg option.>> g++ -pg <<application.cpp>>After the program has been compiled it will output a gmon.out file on run containing the application performance information. This can be viewed in the gprof application.>> gp... | {
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} | 1809.07794 | Evolving system bottlenecks in the as a service cloud | [
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df26d4d6f3a6bcf7cfe682c2dab270874b607b96 | subsection | 24 | 35 | gprof | Detection of independent sub-graphs in the application code is used in application parallelizing. Function call and for loops can be partitioned by the compiler at compile time to optimize the application code for a multi-process / distributed cloud platform. Addition of user directives in the application code enables ... | {
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} | 1809.07794 | Evolving system bottlenecks in the as a service cloud | [
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29682b860e7270963fdb7e5dd7752d6a5d5c6186 | subsection | 25 | 35 | Data storage, retrieval and presentation | We explore the usage of various visualization features in presenting the perf report data to the customer. We use the script command to output a raw report for the performance data. This includes information on the application command, time stamp and shared library related to the event. This data is exported into an Ex... | {
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2318df2d0f343558f9543e2c9be2047c64121d1f | subsection | 26 | 35 | Logstash | Logstash is used to input the user data into Kibana. A conf file is used to setup the Logstash pipeline – Figure REF .
[Figure: Logstash pipeline.]We use the csv input filter to read the csv data file. An output filter is used to add the data to the Elasticsearch index. A template format is specified for reading the c... | {
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11d280bf93f1cb44391b998a0b0033a72ac341b3 | subsection | 27 | 35 | Elasticsearch | Elasticsearch is a distributed and scalable data store. It provides search and analytics capabilities with Rest services. It supports indexing of a JSON document. Input is through Logstash. | {
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970118d81f0e5a76a1036f379886cc3ccb5f701d | subsection | 28 | 35 | Kibana | The data set is loaded in Kibana and a visualization dashboard is created to display the performance data to the customer.Data output in json format
Stored in Kibana Elasticsearch
Input using Logstash>> curl 'localhost:9200/_cat/indices?v'>> curl -XDELETE 'http://localhost:9200/linuxperf'>> curl -XPUT localhost:9200/... | {
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4244d05422e8e9c88991bbe5a67d60def2596304 | subsection | 29 | 35 | Characteristics of the user interface and the details about its functionality | Kibana is an open source framework for data visualization . It enables the user to analyze and explore structured and unstructured data in an intuitive interface. It supports the use of graphs, histogram and pie charts on large amounts of data. Users can view and discover the data in the UI. They can create various vis... | {
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30aa445104d7e3ec1ad6ca7b08366e41c191db01 | subsection | 30 | 35 | Kibana user interface visualization | We evaluate a network workload consisting of a secure file transfer and zip compression of the data. As can be seen in the data a large amount of time is utilized in the libcrypto and gzip application. The histogram plots the total number of events per second as recorded / reported in perf. We additionally output a pie... | {
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74365f819d90662ab0817514eae3a25adecf423f | subsection | 31 | 35 | Python, MongoDB, Tomcat server | We additionally investigate the use of a NoSQL database in storing the results data. This is an architecture deployed in the Holmes Helpdesk platform. Python is used to update the performance profiling data in MongoDB code. This is then output to the customer in a HTML file in a DataTable - Figure REF .
[Figure: HTML ... | {
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1fb3d82205df3204242102b7d9a44d1899285f63 | subsection | 32 | 35 | Functional and nonfunctional requirements of the profiler | The performance profiler has been developed for a Linux platform and supports the Unix POSIX function calls. It should support the C++, Java development libraries and run-time environments. There should be a compatible JRE running on the system. Additionally, a lot of the UI features will be supported in a browser base... | {
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a988c777875da1bbf1309fa8c8268392a2216209 | subsection | 33 | 35 | Functional and nonfunctional requirements of the profiler | The CDT core supports a Visitor API which is used to traverse the AST. AST rewrite API is used to update the source code. We access the code AST using the Eclipse CDT API.We evaluate the generation of a dependency graph for software profiling. The AST is used to profile critical application paths including access to lo... | {
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7781aa6794b46cb8360730c0816552de3574f819 | subsection | 34 | 35 | Conclusion | We have looked at architectures for the next generation enterprise including end to end solutions for the web infrastructure. These highlight the challenges in bringing billions of users online on a commodity platform. There is a large opportunity in enabling technology consumption for more than a billion users. Techno... | {
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138ffd61331446866e0e44cbb4a27f2c209e6726 | abstract | 0 | 76 | Abstract | We define lowest weight polynomials (LWPs), motivated by $so(d,2)$
representation theory, as elements of the polynomial ring over $ d \times n $
variables obeying a system of first and second order partial differential
equations. LWPs invariant under $S_n$ correspond to primary fields in free
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cd536e36d587f071d6619c4d89fe929dbc162de6 | subsection | 1 | 76 | Introduction | The counting and construction of primary fields in free scalar field theories
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General primary fields in scalar field theory in d dimensions, which are composites of
n elementary fields, are in 1-1 correspondence with
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612b385a6887d0563d1f5cca1fa04601c822db39 | subsection | 2 | 76 | Introduction | Our first new result (Section REF ) is to give an explicit description of the quotient ring in dimension d in terms of (n-1)d generators and explicit quadratic relations. The quadratic relations are given in terms of a Clebsch-Gordan decomposition problem for S_n, which we explicitly solve.The Hilbert series of the quo... | {
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ebfd79df6a9b2a7a51ddfeceaa410c344a8fd854 | subsection | 3 | 76 | Introduction | If we denote by V_Q the vector space of quadratic constraints,
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8c1054219bd5a43b44b9f20ba45f57e56e5fd111 | subsection | 4 | 76 | Primary fields from differential constraints and Polynomial Rings | A key result motivating our study is the observation that primary fields constructed from n copies of
\phi , along with their derivatives, correspond to polynomials in variables x_{\mu }^I subject to a system of
linear differential constraints and an S_n invariance condition.
There are d first order differential constr... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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c3cf49e9691048ab654f50059a0825e565777b86 | subsection | 5 | 76 | Review : Lowest weight states and primaries from differential equations | The scalar field and its derivatives form a vector space V, which is an irreducible representation of so(4,2). This representation is isomorphic to the space of harmonic polynomials in x_{\mu }. The connection between the standard action of the conformal group on the fields and the action of differential operators on t... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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c715cc947ff7ab4758ac7dc6a69a11ed33c25a89 | subsection | 6 | 76 | Review : Lowest weight states and primaries from differential equations | Polynomials in x_{ \mu }^I which are harmonic in each of the x_{\mu }^I, i.e. which are annihilated by the n operators\sum _{ \mu } { \partial ^2 \over \partial x^I_{\mu } \partial x_{\mu }^I }span the space \mathcal {H}^{ \otimes n } .A lowest weight polynomial (LWP) denoted L ( x_{ \mu }^I ) satisfies the equations&&... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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6157c7844db93d27a62a3e8b779bcf24c8660c69 | subsection | 7 | 76 | LWPs and the quotient ring | The polynomial ring \mathbb {C}[ x_{\mu }^I ] is denoted as \mathcal {R} .
Consider the ideal \mathcal { I } generated by the n elements \sum _{ \mu }
x^I_{ \mu } x_{ \mu }^I along with the d elements \sum _{ I } x_{\mu }^I.
This is denoted by\mathcal { I } = \langle \sum _{ \mu } x^I_{\mu } x_{ \mu }^I , \sum _{ I } x... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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ba40c9af21e064340965b640bddb8f2cc0e892c0 | subsection | 8 | 76 | LWPs and the quotient ring | Thus, we can write\mathcal {H}^{ \otimes n } = \bigoplus _{ k , j_1 , j_2 } \mathcal {H}_{ n + k , j_1 , j_2 } \otimes \mathcal {M}_{ k , j_1 , j_2 }For classification of the irreps of so(d,2) and their character formulae, see
and refs therein.
A lowest weight state with \Delta = n +k generates a tower of states at hi... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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988095a55669109c74fb462e2c6989908d8f1bc9 | subsection | 9 | 76 | Representation theory of | The I index of x_\mu ^I, ranging over 1 \le I \le n ,
transforms in the natural representation, V_{ {\rm {nat}}} of S_n.
This representation has an orthogonal decomposition into irreducible representationsV_{ {\rm {nat}}} = V_0 \oplus V_HV_0 is the one-dimensional representation. V_H has dimension (n-1)
and corresponds... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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0... |
023038a90084e8cfd5823084db86b836ac22b1c9 | subsection | 10 | 76 | Representation theory of | Introducing the notation S_{ A I} for these
coefficients we havee_A = \sum _{ I =1 }^{ n} S_{A I } e_Ifor A \in \lbrace 1, 2, \cdots , n -1 \rbrace , andS_{ A I } = { 1 \over \sqrt{ A ( A+1)} } \left( - A ~ \delta _{ I , A +1 } + \sum _{ J =1}^{ A } \delta _{ J , I} \right)It is also useful to introduce extend A to A \... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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-... |
80a886ff195bff2635b696f6e30fee7c26dff9d4 | subsection | 11 | 76 | Representation theory of | Note that \kappa _{ABC} has the following S_n invariance property.\kappa _{ABC} &=& \sum _{ I =1}^n \langle H, C | {\rm {nat}}, I \rangle \langle H, B | {\rm {nat}}, I \rangle \langle H, A | {\rm {nat}}, I \rangle \cr &=& \sum _{ I =1}^n \langle H, C | {\rm {nat}}, \sigma ( I ) \rangle \langle H, B | {\rm {nat}}, \sigm... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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... |
2f8522e6f261ce20c90e5b125678c7109b70fd54 | subsection | 12 | 76 | Representation theory of | We also find that\kappa ( z_A ) & \equiv & \sum _{A, B , C =1 }^{ n-1} \kappa _{ ABC} z_A z_B z_C \cr & = & \sum _{ A } A ( 1- A^2) z_A^3 + \sum _{ A < B } 3 A ( 1 + A ) z_A^2 z_Band\kappa _{ A } ( z ) & = & \sum _{ B , C } \kappa _{ A B C } z_B z_C \cr & = & A ( 1 - A^2 ) z_A^2 + \sum _{ B : B < A } B ( 1 + B ) z_B^2
... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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260387b416bf5ad28e98baee4c82405bc3f43eb9 | subsection | 13 | 76 | Solving the center of mass constraint and a polynomial ring with quadratic relations | In solving the constraints that determine the LWPs, a fruitful approach is to solve the
center of mass constraint (COM) and only then consider the remaining constraints in (REF ).
This approach exploits the S_n structure of the problem. We will use the elements
of S_n representation theory from Section REF .As noted ea... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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f57b40fd77a23af7dcbd90b72d2fb6dcfad1ec98 | subsection | 14 | 76 | Solving the center of mass constraint and a polynomial ring with quadratic relations | Given the explicit formulae stated in Section REF and derived
in Appendix , the quadratic constraints
can be expressed as&& \hbox{ For } 1 \le A \le (n-1) : \cr && \cr && A ( 1 - A^2 ) X^{ A }_{ \mu } X^{ A }_{ \mu } + \sum _{ B : B > A } 2 A ( 1+A ) X^{A}_{\mu } X^{ B}_{ \mu } + \sum _{ B : B < A } B ( 1+ B ) X^{B}_{\... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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a80d6b0af41041548e4e16ac072b050531086f87 | subsection | 15 | 76 | Body | The LWPs solve the n Laplacian conditions\sum _{ \mu } { \partial ^2 F \over \partial x_{ \mu }^I \partial x_{ \mu }^I } = 0These n conditions transform in the natural representation V_{nat} of S_n.
We can again move to the V_0 \oplus V_{H} basis as follows\Box _{ C } = \sum _{ I =1 }^n S_{C I } { \partial ^2 \over \pa... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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e25085256aa605fd2fd8197445f3ccd933c708c5 | subsection | 16 | 76 | Counting of lowest weight states in | V is the representation of so(d,2) collecting all the states which correspond, by the operator-state correspondence,
to a single scalar field and its derivatives. Above we have established that the lowest weight states in V^{\otimes n}
form a polynomial ring. We will develop this description further in this section by ... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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6e488c8cea3e1c85ff7977f0c7633fe24709ad82 | subsection | 17 | 76 | Counting of lowest weight states in | An important point
for the discussion in the next section is that{ 1\over ( 1 - s)^{ d ( n -1) } } { n k } s^{ 2k}is the trace of s^{ \Delta } over \mathcal {R} \otimes \Lambda ^k ( V_Q ) .
Finally, it is worth noting that the counting function in the first line of
(REF ) is palindromic. | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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0... |
2abcb58dcedd5bc99221cb20c722dab95b6696e6 | subsection | 18 | 76 | The ring of lowest weights in | In the previous section we have obtained the counting function for the lowest weights in V^{\otimes n}.
These lowest weights form a polynomial ring. The counting function for the ring
is a rational function. The ring is a quotient of the polynomial ring, by an ideal.
The ideal is generated by n quadratic expressions.Th... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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43cd667bbe087305646829ee9231293f3da3fae7 | subsection | 19 | 76 | Exact sequence of modules | We will consider the following exact sequence of modules over \mathcal {R} = \mathbb {C}[ X_{\mu }^{A}]&& 0 \xrightarrow{} \mathcal {R} \otimes \Lambda ^{ n } ( V_Q ) \xrightarrow{} \cdots \rightarrow \mathcal {R} \otimes \Lambda ^2 ( V_Q ) \xrightarrow{} \mathcal {R} \otimes V_Q \xrightarrow{} \mathcal {R} \xrightarro... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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274d8d5ee2e5c72559abf52e7ebf877dd007c485 | subsection | 20 | 76 | Exact sequence of modules | Under the map f_I, they go to\epsilon ^{ A_1, \cdots , A_I, A_{ I+1}, \cdots , A_n } h_{ A_1 A_2 \cdots A_I } Q_{A_1} \otimes Q_{A_2} \cdots \otimes Q_{ A_I}Under the composite map f_{ I } \circ f_{ I-1}, we have&& f_{ I } \circ f_{ I-1} :
\epsilon ^{ A_1, \cdots , A_I, A_{ I+1}, \cdots , A_n } h_{ A_1 A_2 \cdots A_I }... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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... |
836b5990e07e99ced3b012ab939564891f7b5fb8 | subsection | 21 | 76 | Exact sequence of modules | To motivate the second operator we need, we employ a decomposition of polynomials that will be derived in
Section REF : the space of degree k polynomials can be decomposed asp^{(k)} = p_h^{(k)} + Q_A p_{h,A}^{ (k-2) } + Q_AQ_B p_{h,AB}^{(k-4)} + \cdotswith the coefficients p_h^{(k )}, p_{h,A}^{(k-2)}, p_{h,AB}^{(k -4)}... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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-... |
01084a8447ee5ce7bdeac3314d5143ad6dde99d9 | subsection | 22 | 76 | Exact sequence of modules | In fact, we will show that acting on a monomial of degree t in the Qs, it is proportional to the identity\alpha \circ d+d\circ \alpha = 1+tIn this case, acting on any element
k^{(q)}_{h,AB\cdots E}Q_AQ_B\cdots Q_E\otimes Q_{A_1}\wedge \cdots \wedge Q_{A_i} in the kernel of d, we have(\alpha \circ d+d\circ \alpha )k^{(q... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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... |
7aefa1bc46711630d2d3c1649a6f44ee9a32af12 | subsection | 23 | 76 | Exact sequence of modules | For d\circ \alpha we find (assume that p^{(q)}_{h,AB\cdots E} has t indices AB\cdots E)&&d\circ \alpha (p^{(q)}_{h,AB\cdots E}Q_AQ_B\cdots Q_E\otimes Q_{A_1}\wedge \cdots \wedge Q_{A_i})\cr \cr &=& d\left(\sum _{S\notin \lbrace A_1,...,A_i\rbrace }
t p^{(q)}_{h,SB\cdots E}Q_B\cdots Q_E\otimes Q_{A_1}\wedge \cdots \wedg... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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9a3c76abc94f229b13715dc238166be1f7de0767 | subsection | 24 | 76 | Exact sequence of modules | One is the standard short exact sequence for quotients0 \rightarrow \mathcal { I } \rightarrow \mathcal {R} \rightarrow \mathcal {R} / \mathcal { I } \rightarrow 0The next isSyz ( \mathcal { I } ) \rightarrow \mathcal {R} \otimes V_Q \rightarrow \mathcal { I }Here a basis for V_Q gives the generators of \mathcal { I } ... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
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9d38c9a25a48193b675e3cbeff29be6b7eb3165a | subsection | 25 | 76 | Exact sequence of vector spaces over | We can consider the vector space formed by polynomials of a fixed degree.
Since the modules of the last section are defined over the ring, a single exact sequence of modules implies, upon specializing to fixed degree, an exact sequence for each of these vector spaces.
The polynomials at fixed n and fixed degree k are p... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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81baca0d68ec5960e0ad3b766b61f90bd0085de2 | subsection | 26 | 76 | Exact sequence of vector spaces over | Start by introducing the map f defined byf : \hbox{Sym} ^2 ( V_{dH} ) \otimes V_Q \rightarrow \hbox{Sym} ^4 ( V_{dH} )Concretely, we havef : Q_A \otimes X_{ \mu _1}^{ a_1} X_{ \mu _2}^{ a_2 } \rightarrow Q_{ A} X_{ \mu _1}^{ a_1} X_{ \mu _1}^{ a_2 } = \kappa _{A B C } X_{ \mu }^{ B } X_{ \mu }^{ C }X_{ \mu _1}^{ a_1} X... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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b7b7f191c63d797257fc129a21e074664b52e490 | subsection | 27 | 76 | Exact sequence of vector spaces over | If n \le (k-1)/2,
then it is \hbox{Sym} ^{ k - 2n } \otimes \Lambda ^{ n } ( V_Q) .One basic building block that the above sequences are constructed from is the following\cdots \rightarrow \hbox{Sym} ^{k-2I}(V_{dH})\otimes \Lambda ^{I}(V_Q)
\rightarrow \hbox{Sym} ^{k+2-2I}(V_{dH})\otimes \Lambda ^{I-1}(V_Q)\cr \rightar... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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5046da557e5098bbb1b6d8895a49cd7d52bacd95 | subsection | 28 | 76 | Exact sequence of vector spaces over | Indeed, the image of g is spanned byX^{a_1}_{\mu _1}\cdots X^{a_{k-2I}}_{\mu _{k-2I}}\kappa _{A_{I}BC}X^B_\mu X^C_\mu \otimes \epsilon ^{A_1\cdots A_{I-1}A_I\cdots A_L}Q_{A_1}\otimes Q_{A_2}\otimes \cdots \otimes Q_{A_{I-1}}Under f this maps to zero&&f(X^{a_1}_{\mu _1}\cdots X^{a_{k-2I}}_{\mu _{k-2I}}\kappa _{A_{I}BC}X... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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18cb3b0f0392b797f85dbf61df08cd48046ddb2c | subsection | 29 | 76 | Exact sequence of vector spaces over | The generalization of the argument is then obvious and we will not repeat it here.The exact sequences we have presented in this section imply thatL(k,d,n) = \sum _{I=0}^{min ( \lfloor {k\over 2}\rfloor , n ) }(-1)^I \hbox{Dim}( \hbox{Sym} ^{k-2I}(V_{dH})) \hbox{Dim}(\Lambda ^I (V_Q))This formula will be used in the nex... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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4ca479e6270c10ceff3e8cd3c70e053e15cff914 | subsection | 30 | 76 | Refined counting formulae | We have managed to count the number of LWPs of fixed degree, or equivalently,
lowest weight states in V^{ \otimes n }.
There are good reasons to refine this counting using the so(d) \times S_n symmetry present in the problem. Primaries in the free field theory are S_n invariants. They
are labeled by their dimension and... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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0.... |
dcf1b26ea59991abb7473ed4d34099ce414a68ac | subsection | 31 | 76 | Refined counting formulae | We can now write the so(d) \times S_n refined version of (REF ) as&& L ( \Lambda _1^{ so(d)} , \Lambda _3^{ (S_n)} ; k , d , n ) \cr &&
= \sum _{ \Lambda _{3 , 1 }, \Lambda _{ 3,2} \vdash n } \sum _{ I = 0 }^{ min ( \lfloor {k\over 2}\rfloor , n ) } (-1)^I {\rm Mult} ( ( \hbox{Sym} ^{k - 2I } ( V_{ dH } ) ; \Lambda _1^... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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36c1297f13cf6c1ad646e0b4953649006d474c50 | subsection | 32 | 76 | Refined counting formulae | The result is&& {\rm Mult} (\Lambda ^I (V_{nat}^{(S_n)});\Lambda _{3,2}^{(S_n)})\cr &&=\sum _{p\vdash n}\sum _{q\vdash I} {(-1)^{q_2+q_4+\cdots }} {\chi _{\Lambda _{3,2}}^{p}\over Sym(p) Sym(q) }\prod _{i=1}^I \left(\sum _{d|i} d p_d \right)^{ q_i }where \chi _{\Lambda _{3,2}} is the S_n character of the permutation wi... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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ed078b3312195f8f5c1294072c04ba2ce6244441 | subsection | 33 | 76 | Refined Counting: general | We can make the formula (REF ) more explicit for general d.
The formulae we write here will be not be as computationally efficient as for d=3,4
but may still be useful in further studies of so(d,2) representations and free field primaries
in higher dimensions.
We focus on the d-dependent quantity{\rm Mult} ( \hbox{Sym}... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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b818cdb7b5d33a81b159d2b045fcae9df2e8eeea | subsection | 34 | 76 | Refined Counting: general | This leads to&& {\rm Mult} ( \hbox{Sym} ^k ( V_{dH } , \Lambda _1^{ so(d)} \otimes \Lambda _3^{ (S_n)} )
= \sum _{ \Lambda _2 \vdash k } {\rm Mult} ( V_d^{ \otimes k } , \Lambda _1^{ so(d) } \otimes \Lambda _2^{ (S_k)} )
{\rm Mult} ( V_H^{ \otimes k } , \Lambda _3^{ (S_n)} \otimes \Lambda _2^{ (S_k) } ) \cr &&The {\rm... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
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db1b81ef88052aa6f230e71e4c31ad9a9217ac59 | subsection | 35 | 76 | Refined counting: | We need the multiplicitites of V_{ \Lambda _1^{so(3)}}\otimes V_{\Lambda _2^{(s_k)}} in V_{3}^{\otimes k}.
This problem has been considered in . Our coordinates X_\mu ^A are in the 3 of so(3), which is the spin 1 of SU(2). For d=3, \Lambda _1 is parameterised by one integer l for the spin.
Our multiplicities are given ... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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3a0642ba4fbae5503490d3750d09542c32c1d61e | subsection | 36 | 76 | Refined counting: | The outcome is&& {\rm Mult} ( \hbox{Sym} ^{k - 2I } ( V_{dH}) ; ( \Lambda _1^{ so(d)} = [ l ] ) \otimes \Lambda _{3,1}^{ (S_n)} ) ) \cr && = \sum _{ \Lambda _2 \vdash k - 2I }
{\rm Mult} ( V_{ 3}^{ \otimes k - 2 I } , [ l ] \otimes \Lambda _2 ) )
{\rm Mult} ( V_{ H }^{ \otimes k - 2I } , V_{ \Lambda _{ 3,1} }^{ ( S_n... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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9bff528252bf2ea734ddae45b85fa7b8aa220f20 | subsection | 37 | 76 | Refined counting : the | The fundamental of SO(4) is the V_{1/2}\otimes \bar{V}_{1/2} of SU_L(2) \times SU_R(2), where V_{1/2} is the two-dimensional spin half irrep of SU(2). X_{ \mu }^{ A} transforming
in V_d \otimes V_H can be written as X_{ \alpha , \dot{\alpha }}^A to reflect the description as
V_{1/2} \otimes V_{ 1/2} \otimes V_d. It is... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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c056bcd17b14b87a9965e10a56996a3aaa3ca7a3 | subsection | 38 | 76 | Refined counting : the | The final result is&& \hbox{Sym} ^k ( V_{ dH} )
= \bigoplus _{{ \Lambda _3 \in Y^{(n)} \\ \Lambda _{ 2,1} \in Y_{ 2}^{ (k) } \\
\Lambda _{ 2,1} \in Y_{ 2}^{ (k) } }} V_{ \Lambda _{2,1} }^{ u_L(2)} \otimes V_{ \Lambda _{2,2} }^{ u_R (2)} \otimes V_{ \Lambda _3}^{ ( S_n)} \otimes V_{ \Lambda _3 , \Lambda _{ 2,3} } \otime... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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eda0dfaa2cbc602803460b482fbd49cf0cf2c565 | subsection | 39 | 76 | The Confluent Binomial Transform and construction | The exact sequences we derived in Section have led to a dimension formula for \mathcal {L}(k,d,n) (or for \mathcal { I } (k,d,n)) as an alternating sum involving exterior powers of V_Q. In this section we will show that there is a dimension formula for \mathcal { I } (k,d,n) as a positive sum involving symmetric powers... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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87a8a3b77097f9f7042b1df9f9a530243f0ed324 | subsection | 40 | 76 | From resolution to construction : counting without signs | Using characters we have obtained the generating function of the number of LWPs as follows{(1-s^2)^n\over (1-s)^{d(n-1)}} = \sum _{l =0}^{ \infty } L(l,d,n) s^{ l }We will derive an interesting expression for L(l,d,n) in terms of S(k,d,n) which is the dimension of \hbox{Sym} ^k (V_{dH}).
Our starting point is the expli... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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f7ca61de52e8efe8db5f08a470e255dd80b91c22 | subsection | 41 | 76 | From resolution to construction : counting without signs | ( n-k) ! } S ( p - 2m , d, n )This is indeed an equality, which follows after using the identity\sum _{m =0}^{\lfloor {p\over 2}\rfloor }\sum _{k=0}^{\min (m,n)}{(-1)^k \over (m-k)! k!(n-k)!}={1\over n}\delta _{m,0}The equation (REF ) implies, after subtracting the i=0 term L ( p , d, n ) ,
that the dimension of the id... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
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0.029792545363307,
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0.006895150989294052,
0.017649756744503975,
-0.03374352678656578,
0.037984348833560944,
-0.023614367470145226,
-0.00... |
e19d53d294dd91f09de787826e19f37f285993cc | subsection | 42 | 76 | From resolution to construction : counting without signs | Consequently,
a subspace of \mathcal { I } ( k , d , n ) is\mathcal {L}( k -2 , d , n ) \otimes V_QAt this point, we use the identity (), derived with the help of the confluent binomial transform, to decompose
\mathcal { I } (k,d,n) as\mathcal { I } ( k , d , n ) = \bigoplus _{ i =1 }^{ \lfloor { k \over 2} \rfloor } ... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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... |
2848bc7cd426e48e695e25a4eba1f1aedc12f5e9 | subsection | 43 | 76 | From resolution to construction : counting without signs | The symmetric algebra of V_Q, denoted by \hbox{Sym} ( V_Q ) is the direct sum
of symmetrised tensor products of all degrees\hbox{Sym} ( V_Q ) = \bigoplus _{ k=0}^{ \infty } \hbox{Sym} ^k ( V_Q )The degree 0 part is defined as \mathbb {C}. The above decompositions of \mathcal {R} at each degree are captured by\mathcal {... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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aca96022b68c728811792472cfb21348f1580d54 | subsection | 44 | 76 | Implementing the construction using the natural inner product | In the last section we have described an algorithm for the construction of the polynomials that correspond to primary operators. The algorithm works by recursively proceeding in degree, using orthogonality to construct the higher degree spaces from the lower degree ones. The only missing ingredient in the algorithm was... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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... |
665182beba6fa1cfd8d2a8651c2077d6e1af19e1 | subsection | 45 | 76 | Implementing the construction using the natural inner product | The traceless tensors correspond to harmonic polynomials, which establishes the result.For simplicity, start with the single particle case.
Consider the differential operator
\sum _{\mu ,\nu =1}^d x_{\mu } x_{\mu }{\partial ^2 \over \partial x_{\nu }\partial x_{\nu }}. Let it act on all polynomials of fixed degree - it... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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f1c7a71b29ac8bb91f8cfacb53da6e57ffdda4e8 | subsection | 46 | 76 | Implementing the construction using the natural inner product | A second approach to demonstrate the same fact, makes use of the operator\mathcal {O} _1 =\sum _{I,J=1}^n x^I{\partial \over \partial x^J}which is hermitian with respect to the natural inner product.
The translation invariant polynomials belong to the null space of \mathcal {O} _1, while the non-translation
invariant e... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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f98a92a1dbf7a5f3c6cd99d9794111939c089a00 | subsection | 47 | 76 | Implementing the construction using the natural inner product | The above discussion implies that the space of degree k polynomials can be decomposed asp^{(k)} = p_h^{(k)} + Q_A p_{h,A}^{ (k-2) } + Q_AQ_B p_{h,AB}^{(k-4)} + \cdotsp_h^{(k)},p_{h,A}^{(k)},\cdots are all polynomials of degree k which are annihilated by the
Laplacians \Box _A defined in (REF ).
In the expansion only th... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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4c3265b31a6caf5bf0032fddd40b6ea53a810b04 | subsection | 48 | 76 | Commutative star product on lowest weight polynomials | As we saw in Section the lowest weight polynomials are in 1-1 correspondence with a quotient ring, which
has an associative product inherited from the quotient construction.
Since the Laplacian constraints obeyed by the polynomials of the ring are second order differential operators, given two polynomials that obey the... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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... |
7e67c78e7e283884562f4892f5e22d3d3be1bd8a | subsection | 49 | 76 | Commutative star product on lowest weight polynomials | Using (REF ) decompose the product of two polynomials, which each obey the Laplacian constraintsf^{(k_1)}_h g^{(k_2)}_h&=&(fg)^{(k_1+k_2)}_h+Q_C (fg)^{k_1+k_2-2}_{h,C}+\cdots \cr \cr &+&Q_{C_1}\cdots Q_{C_m}(fg)^{(k_1+k_2-2m)}_{C_1\cdots C_m}The star product we want isf^{(k_1)}_h * g^{(k_2)}_h=(fg)^{(k_1+k_2)}_hOnly th... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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-... |
5dbf0e746d1ede7774022caf31d679e560251120 | subsection | 50 | 76 | Commutative star product on lowest weight polynomials | Recall that the usual product on polynomials is associativef^{(k_1)}_h(g^{(k_2)}_h h^{(k_3)}_h)=(f^{(k_1)}_hg^{(k_2)}_h)h^{(k_3)}_hRefining both sides of this last equation according to degree, we can write this as&& f^{(k_1)}_h \left((gh)^{(k_2+k_3)}_h+Q_A (g h)^{(k_2+k_3-2)}_{h,A}+\dots \right) \cr &&\qquad \qquad =(... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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-0.... |
3d0890e0b937cbabdf70db1b51860bd56bd8b61c | subsection | 51 | 76 | Further construction methods for lowest weight polynomials | The construction algorithm we gave in the previous section
has a recursive nature, and produces all the LWPs of degree up to any chosen maximum k.
At each k, it uses orthogonality to elements written in terms of the LWPs at lower k.
A second, more direct, algorithm
works at fixed k, and implements the differential equa... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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0.024... |
f95c6ec2fa4be5959a152570b6a9ec94d3ac9b9b | subsection | 52 | 76 | Intersection of Kernels of two differential operators | In this section we will outline a closely related but distinct construction algorithm. This new
algorithm uses the fact that, as we explained earlier, the space of LWPs can be identified as the common null space of
a set of differential operators. Here we will consider degree preserving version of the differential oper... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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... |
82dadde3f7188bae27fca5fc6f5147839dcfb362 | subsection | 53 | 76 | Intersection of Kernels of two differential operators | (d-1)!}\mathcal {W}_{k;n,d} = \bigoplus _{ { k_1 , k_2 \cdots , k_n =0 \\ \sum _{ I } k_I = k } }^{ d} \hbox{Sym} ^{ k_1 } ( V_d ) \otimes \hbox{Sym} ^{ k_2 } ( V_d ) \otimes \cdots \otimes \hbox{Sym} ^{ k_n } ( V_d )On this subspace we have the linear operators\mathcal {O} ^{ (I)}_{ \mathcal {L}} = ( x^{I})^2 \Box _{(... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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... |
410401f007860d9b032e26dd724c79e6f15b2cd4 | subsection | 54 | 76 | Intersection of Kernels of two differential operators | Therefore the following sum\mathcal {O} _{ \mathcal {C} \mathcal {M} } && = \sum _{ \alpha =1 }^{d} x^{ CM}_{ \alpha } { \partial \over \partial x^{ CM}_{ \alpha } } \cr && = \sum _{ I , J =1}^{ n } \sum _{ \alpha =1 }^{d} x^{I}_{ \alpha } { \partial \over \partial x^{J}_{ \alpha } }has the property that its null space... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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0.005839... |
8a92cdd7ac0572c878db2b51dc1ebbb6853e4600 | subsection | 55 | 76 | Constraints and Projectors on | An interesting algebraic angle on the primaries problem is that in some sense it is a generalization of the problem of finding symmetric traceless tensors of SO(d).
Nice bases for these tensors can be constructed using Young diagram techniques for SO groups.
These symmetric traceless tensors are annihilated by contract... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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400cc31dd8d499630e8077eb49bc2f329b1e08e3 | subsection | 56 | 76 | Constraints and Projectors on | So we are looking at vectorsv \in \hbox{Sym} ^2 ( V_{dH} )which obey( P^{ (SO(d)}_{ 0 } ( P_{0 }^{ (S_n )} + P_{ H}^{ (S_n)} ) ) ~~ v = 0Let us defineP^{ \mathcal {L}} = ( P^{ (SO(d)) }_{ 0 } ( P_{0 }^{ (S_n )} + P_{ H}^{ (S_n)} ) )The operator P^{ \mathcal {L}} is a projector obeying (P^{ \mathcal {L}}) ^2 = P^{ \math... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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a9a803fcc93e8b3bdbc6159531b5e2caabf24c8d | subsection | 57 | 76 | Constraints and Projectors on | We will now argue that to find a linear basis at degree k in the ring of LWPs, we have to consider
symmetric tensors T obeying equation (REF ).Use the inner product for polynomials in x_{ \mu }^I used before.
This induces an inner product of the same form on X_{ \mu }^A.
For operator\Box _A = \sum _{ \mu =1 }^d \sum _{... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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-0.... |
4d95c85c610c56a7248d0f81c8f6ffe19648a757 | subsection | 58 | 76 | Standard algebraic geometry methods for | As explained in Section , the LWPs are
in 1-1 correspondence with the elements of \mathcal {R} / \mathcal { I } .
The quotient ring is defined in terms of equivalence classes. Each equivalence class contains
an LWP. There are standard algebraic geometry methods, based on Groebner bases,
for the construction of the equi... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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-0.016678614541888237,... |
9e33f36e712bb382a34e253176cc9a686aa2880f | subsection | 59 | 76 | Discussion and Future Directions. | We have considered the problem of constructing primary fields in free scalar CFTs in general dimensions,
combining insights from , , and .
This has been a fruitful avenue, with the key results
described in the introduction and developed in the bulk of the paper.A number of future projects are suggested by our results.
... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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e2f9ef911c0cff74d82905ba4779f6dc5cc87067 | subsection | 60 | 76 | Further developing the analogy to tracelessness : a generalization of Brauer algebras | In section REF we developed an approach to the construction of
LWPs, based on projectors acting on degree k polynomials
in X_{ \mu }^A. These polynomials form a vector space isomorphic to the space of
symmetric tensors \hbox{Sym} ^k ( V_{ dH} ) .
It is useful to consider the tensor product V_{ dH}^{ \otimes k }
where ... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
"math.RT"
] | 2,018 | en | Physics | [
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... |
3be38146977fb9622af721bfa679bc352a6001af | subsection | 61 | 76 | Quadratic algebras and Koszul algebras. | We have shown that the LWPs are in 1-1 correspondence
with the quotient ring \mathcal {R} / \mathcal { I } , where \mathcal {R} = \mathbb {C}[ X_{ \mu }^A ]
and \mathcal { I } is generated by (REF ).
This is an example of a quadratic algebra. These are defined by quotients of
the tensor algebra T ( \mathbb { V } )
of... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
"hep-th",
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eebcac64fd3bb85981c1fe1df8791da0cf42542a | subsection | 62 | 76 | Quadratic algebras and Koszul algebras. | An important observation is that T ( \mathbb {V} ) / < R >
is a commutative algebra due to the presence of \Lambda ^{ 2 } ( V_{ dH } )
as a direct summand in R, while T ( \mathbb {V} ) / < R^{ \perp } > is not commutative
due to the lack of such a direct summand.A special class of quadratic algebras are said to be Ko... | {
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} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
"Sanjaye Ramgoolam"
] | [
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8178f5daaa10de6954564be18964dddfdeb83129 | subsection | 63 | 76 | Future direction : Coherence relations between two products. | We showed in that the OPE in free scalar theory can be used to
define a commutative so(4,2) covariant algebra with a non-degenerate bilinear pairing. The crossing equation of CFT
becomes ordinary associativity of the algebra. Here we have seen
that there is an algebra controlling primary fields for every n.
The interpl... | {
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} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
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a827d974eb84d01a770aa5a08db6ee8a118cdb29 | subsection | 64 | 76 | The invariant in | In this section we will give the derivation of (REF ).
Inserting the explicit expressions for the S_{AI} we have&& \sum _{ I } S_{ C I } S_{ B I } S_{ A I } = \mathcal {N}_{ A } \mathcal {N}_{ B } \mathcal {N}_{ C }
\sum _{ I } \left( - C ~ \delta _{ I , C +1 } + \sum _{ J_1 =1}^{ C } \delta _{ J_1 , I} \right)
\cr && ... | {
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} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
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32a6552c35a2cc005ca618eb1b9356c2b76c0356 | subsection | 65 | 76 | The invariant in | Define\Theta ( B > C ) && = 1 ~~~ \hbox{ if } B > C \cr && = 0 ~~~~ \hbox{ otherwise }We can then writeT_3 = AC \delta _{ A ,C } \Theta ( B > C )For the fourth term, we haveT_4 && = \sum _{ I =1}^{ n } \delta _{ I , C+1 } \sum _{ J_2 = 1 }^B
\sum _{ J_3 =1 }^{ A } \delta _{ J_2 , I } \delta _{ J_3 , I } \cr && = \sum _... | {
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} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
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34e9d6809937012dffdf38dbbd443ed4caa70b71 | subsection | 66 | 76 | The | The 3-index invariant \kappa _{ ABC } can be used to define a symmetric polynomial in
z_1 , z_2 , \cdots , z_{ n -1} .&& \kappa ( z_1 , z_2 , \cdots , z_{ n-1} )
= \sum _{ A, B , C } \kappa _{ A , B , C } z_A z_B z_C \cr && = - \sum _{ A } A^3 z_A^3 + \sum _{ A > B } B^2 z_A z_B^2 + \sum _{ C > A } A^2 z_A^2 z_C
+ \su... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
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42356d564a3fac7cb0cf323290835445910e8ccf | subsection | 67 | 76 | The | The third term can be manipulated by separated into the case B = C , the case A < B < C and the
case A < C < B to give&& - 3 \sum _{ A < B < C } A z_A z_B z_C =
- 3 \sum _{ A < B } A z_A z_B^2 - 3 \sum _{ A < B < C } A z_A z_B z_C - 3 \sum _{ A < C < B }
A z_A z_B z_C \cr && = - 3 \sum _{ A < B } A z_A z_B^2 - 6 \sum _... | {
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} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
"Robert de Mello Koch",
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69f631fa591bb997b4a86052f733f1d919527a9a | subsection | 68 | 76 | Examples of | Recall that V is the representation of so(4,2) that has all the states which correspond, by the
operator-state correspondence, to the fundamental field and its derivatives.
The unrefined generating function for the fundamental field of so(4,2) is
(the factor in front of the trace below removes the contribution from the... | {
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} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
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87f847b814c70f1e7327c5d6c1340d749f2ea6f0 | subsection | 69 | 76 | Examples of | For example, if n=4 we have the following four constraints (we have rescaled X^{(3)} by \sqrt{2})X^{(1)}\cdot X^{(2)}+X^{(1)}\cdot X^{\prime (3)}&=&0\cr X^{(1)}\cdot X^{(1)}-X^{(2)}X^{(2)}+X^{(2)}\cdot X^{\prime (3)}&=&0\cr X^{(1)}\cdot X^{(1)}+X^{(2)}\cdot X^{(2)}-4X^{\prime (3)}\cdot X^{\prime (3)}&=&0\cr X^{(1)}\cdo... | {
"cite_spans": []
} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
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6d0c3001c6d8c265f46dd7a7095a82002493b978 | subsection | 70 | 76 | Examples of | We have(seefor example \cite{BHR2} for further explanation of this formula)\bea&& Dim ( V_{ \Lambda_2 , \Lambda_1 } ) = Mult ( V_H^{ \otimes k } , V_{ \Lambda_1}^{ S_n } \otimes V_{\Lambda_2}^{ S_k } ) \cr&& = { 1 \over n! k! } \sum_{ \sigma \in S_n } \sum_{ \tau \in S_k } \chi_{ \Lambda_1 } ( \sigma )\chi_{ \Lambda_... | {
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} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
with rings and modules | [
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54bb078524e841692e93b25209ec7fb59ad27589 | subsection | 71 | 76 | Examples of | Now note that we have\beatr_{V_H} ( \sigma ) && = tr_{ nat } ( \sigma ) - tr_{ triv } ( \sigma ) \cr&& = C_1 ( \sigma ) - 1\eeaand\bea&& tr_H ( \sigma^i ) = C_1 ( \sigma^i ) - 1 \cr&& = -1 + \sum_{ d | i } d C_d ( \sigma )\eeaWhen we raise a permutation to power $i$, all cycles of length $d$ which divide $i$spli... | {
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} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
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62443c8b8648b81378446bcad5b4325a6a5aed97 | subsection | 72 | 76 | Examples of | These primaries transformin the spin $(l_1,l_2)$ representation of $so(4)$ and in the $\Lambda_n$ of $S_n$.\begin{table}[H]\center\begin{tabular}{|c|c|c|c|}\hline$l_1$ & $l_2$ & $\Lambda_3$ & Mult\\\hline1 & 1 & [3] & 1\\\hline1 & 1 & [2, 1] & 1\\\hline0 & 1 & [1, 1, 1] & 1\\\hline1 & 0 & [1, 1, 1] & 1\\\hline\end{tabu... | {
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d8681a1fe8fa48e56d77208535a24cb0a6ea9168 | subsection | 73 | 76 | Examples of | A. Cox, J. B. Little and D. O'Shea,``Ideals, Varieties, and Algorithms'' Fourth Edition, 2015, Springer.\bibitem{Eisenbud}Eisenbud, David (1995), Commutative algebra. With a view toward algebraic geometry, Graduate Texts in Mathematics,150, New York: Springer-Verlag, ISBN 0-387-94268-8.\bibitem{Grigorescu}E. Grigorescu... | {
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} | 10.1007/JHEP08(2018)088 | 1806.01085 | Free field primaries in general dimensions: Counting and construction
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99de8fc1b3ecef032f5be2c5d98caeb359648d1d | subsection | 74 | 76 | Examples of | 1, 1985.\bibitem{cjr}S.~Corley, A.~Jevicki and S.~Ramgoolam,``Exact correlators of giant gravitons from dual N=4 SYM theory,''Adv.\ Theor.\ Math.\ Phys.\ {\bf 5} (2002) 809doi:10.4310/ATMP.2001.v5.n4.a6[hep-th/0111222].%%CITATION = doi:10.4310/ATMP.2001.v5.n4.a6;%%%397 citations counted in INSPIRE as of 03 Jun 2018\bi... | {
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8b0179b1854debbc894b6847af92d3905925c56a | subsection | 75 | 76 | Examples of | Polischchuk, L. Positselski, ``Quadratic Algebras'', University Lecture Series 37, Providence, R.I.American Mathematical Society.\bibitem{Wiki-Koszul}Wikipedia article on Koszul duality:\\https://en.wikipedia.org/wiki/Koszul\_duality\bibitem{Szabo}R.~J.~Szabo,``Quantum field theory on noncommutative spaces,''Phys.\ Rep... | {
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e49bde553616c7dbfbd416dfae382c2c1d275c97 | abstract | 0 | 17 | Abstract | In low-rank tensor completion tasks, due to the underlying multiple
large-scale singular value decomposition (SVD) operations and rank selection
problem of the traditional methods, they suffer from high computational cost
and high sensitivity of model complexity. In this paper, taking advantages of
high compressibility... | {
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} | 1805.08468 | Rank Minimization on Tensor Ring: A New Paradigm in Scalable Tensor
Decomposition and Completion | [
"Longhao Yuan",
"Chao Li",
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"Jianting Cao",
"Qibin Zhao"
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d67fe8549fd8c2602d82b7b7752fb1251c4d95bc | subsection | 1 | 17 | Introduction | Tensor decomposition aims to find the latent factors of tensor valued data (i.e. the generalization of multi-dimensional arrays), thereby casting large-scale tensors into a multilinear tensor space of low-dimensionality (very few degree of freedom designated by the rank). Tensor factors can then be considered as latent... | {
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2829c6355062513fa069976a4bd095cf173f2218 | subsection | 2 | 17 | Introduction | However, the two methods need to perform multiple SVD operations on the matricization of tensors, and the computational complexity grows exponentially with tensor dimension. Other tensor completion algorithms, like alternating least squares (ALS) , and gradient-based algorithms , , need to specify the rank of the decom... | {
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e18ef3bae136d6616b2b2ba48ab778c8704083fa | subsection | 3 | 17 | tensor ring decomposition | tensor ring (TR) decomposition is a more general decomposition than tensor-train (TT) decomposition, and it represents a tensor with large dimension by circular multilinear products over a sequence of low dimension cores. All of the cores corresponding to TR decomposition are order-three tensors, and are denoted by {\m... | {
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91dee911bac6be1a1045c6cc746bfcf2dcd8e22b | subsection | 4 | 17 | tensor ring decomposition | \Gamma _n(\cdot ) is the mode-n matricization operator of a tensor, i.e., if {\mathcal {X}}\in \mathbb {R}^{I_1\times I_2\times \cdots \times I_N}, then \Gamma _n(\mathbf {X})\in \mathbb {R}^{I_n \times {I_1 \cdots I_{n-1} I_{n+1} \cdots I_N}}. \Delta _n(\cdot ) is another type of mode-n matricization operator of a ten... | {
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52e4677afa33a9507875ae1bd0016e462c98d647 | subsection | 5 | 17 | Tensor completion by Schatten norm regularization | The low-rank tensor completion problem can be formulated as:\min \limits _{{\mathcal {X}}} \ \ \text{Rank}({\mathcal {X}}),\ s.t. \ P_{\Omega }({\mathcal {X}})=P_{\Omega }({\mathcal {T}}),and the model can be written in a unconstrained form by:\min \limits _{{\mathcal {X}}} \ \ \text{Rank}({\mathcal {X}})+\frac{1}{\lam... | {
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0d526afca885ae610d345c8169776024174a6896 | subsection | 6 | 17 | Tensor completion by tensor decomposition | Some other existing tensor completion algorithms do not employ a low-rank constraint to the tensor, and thus they do not find the low-rank tensor directly, instead, they try to find the low-rank representation (i.e. tensor factors) of the incomplete data by observed entries, then the obtained latent factors are used to... | {
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