Datasets:

ACDRepo
/

schubert_polynomial_structure_constants_5

@@ -19,9 +19,10 @@ their product can be written as a linear combination of Schubert polynomials
 Where the sum runs over permutations in \\(S_{n+m}\\). The question is whether the
 \\(c^{\gamma}_{\alpha \beta}\\) (the *structure constants*) have a combinatorial interpretation.
 To give an example of what we mean by combinatorial interpretation, when Schur polynomials
-(which can be viewed as a specific case of Schubert polynomials) are multiplied together,
 the coefficients in the resulting product are equal to the number of semistandard tableaux
-satisfying certain properties.
 ## Example
@@ -32,7 +33,8 @@ and \\(\mathfrak{S}_{\beta} = x_1\\). Multiplying these together we get
 \\(\mathfrak{S}_{\alpha}\mathfrak{S}_{\beta} = x_1^2 + x_1x_2\\). As
 \\(\mathfrak{S}_{2 3 1} = x_1x_2\\) and \\(\mathfrak{S}_{3 1 2} = x_1^2\\) we can write
 \\(\mathfrak{S}_{\alpha}\mathfrak{S}_{\beta} = \mathfrak{S}_{2 3 1} + \mathfrak{S}_{3 1 2}\\).
-It follows that \\(c_{\alpha,\beta}^{\gamma} = 1\\) if \\(\gamma = 2 3 1\\) or \\(\gamma = 3 1 2\\)
 and otherwise \\(c_{\alpha,\beta}^{\gamma} = 0\\).
 ## Dataset
@@ -50,14 +52,6 @@ of non-zero coefficients).
 **Statistics**
-All structure constants in this case are either 0 or 1 (so the classification problem is binary).
-|  | 0 | 1 | Total number of instances |
-|----------|----------|----------|----------|
-| Train | 851 | 833 | 1,684 |
-| Test  | 201 | 220 | 421 |
 All structure constants in this case are either 0, 1, or 2.
 |  | 0 | 1 | 2 |  Total number of instances |
@@ -65,32 +59,24 @@ All structure constants in this case are either 0, 1, or 2.
 | Train | 42,831 | 42,619 | 170 | 85,620 |
 | Test  | 10,681 | 10,680 | 44 | 21,405 |
-All structure constants in this case are either 0, 1, 2, 3, 4, or 5.
-|  | 0 | 1 | 2 | 3 | 4 | 5 |  Total number of instances |
-|----------|----------|----------|----------|----------|----------|----------|----------|
-| Train | 4,202,040 | 4,093,033 | 109,217 | 2,262 | 9 | 9 | 8,406,564 |
-| Test  | 1,052,062 | 1,021,898 | 27,110 | 568 | 3 | 0 | 2,101,641 |
 ## Data generation
-The Sage notebook within this directory gives the code used to generate these datasets.
 The process involves:
 - For a chosen \\(n\\), compute the products \\(\mathfrak{S}_{\alpha} \mathfrak{S}_{\beta}\\) for \\(\alpha,\beta \in S_n\\).
-- For each of these pairs, extract and add to the dataset all non-zero structure constants \\(c^{\gamma_1}_{\alpha,\beta}, \dots, c^{\gamma_k}_{\alpha,\beta}\\)
 - Furthermore, for each \\(c^{\gamma_i}_{\alpha,\beta} \neq 0\\), randomly permute \\(\gamma_i \mapsto \gamma_i'\\) to find \\(c^{\gamma_i'}_{\alpha,\beta} = 0\\) and \\(c^{\gamma_i'}_{\alpha,\beta}\\) is not already in the dataset.
 ## Task
 **Math question:** Find a combinatorial interpretation of the structure constants \\(c_{\alpha,\beta}^\gamma\\)
-based on properties of \\(\alpha\\), \\(\beta\\), and \\(\gamma\\).
 **Narrow ML task:** Train a model that, given three permutations \\(\alpha, \beta, \gamma\\), can
-predict the associated structure constant \\(c^{\gamma}_{\alpha,\beta}\\).
 ## Small model performance
 | Size | Logistic regression | MLP | Transformer | Guessing majority class |
 |----------|----------|-----------|------------|------------|

 Where the sum runs over permutations in \\(S_{n+m}\\). The question is whether the
 \\(c^{\gamma}_{\alpha \beta}\\) (the *structure constants*) have a combinatorial interpretation.
 To give an example of what we mean by combinatorial interpretation, when Schur polynomials
+(which are a subset of Schubert polynomials) are multiplied together,
 the coefficients in the resulting product are equal to the number of semistandard tableaux
+satisfying certain properties (this is known as the
+[Littlewood-Richardson rule](https://en.wikipedia.org/wiki/Littlewood%E2%80%93Richardson_rule)).
 ## Example
 \\(\mathfrak{S}_{\alpha}\mathfrak{S}_{\beta} = x_1^2 + x_1x_2\\). As
 \\(\mathfrak{S}_{2 3 1} = x_1x_2\\) and \\(\mathfrak{S}_{3 1 2} = x_1^2\\) we can write
 \\(\mathfrak{S}_{\alpha}\mathfrak{S}_{\beta} = \mathfrak{S}_{2 3 1} + \mathfrak{S}_{3 1 2}\\).
+It follows that for these \\(\alpha\\) and \\(\beta\\), \\(c_{\alpha,\beta}^{\gamma} = 1\\)
+if \\(\gamma = 2 3 1\\) or \\(\gamma = 3 1 2\\)
 and otherwise \\(c_{\alpha,\beta}^{\gamma} = 0\\).
 ## Dataset
 **Statistics**
 All structure constants in this case are either 0, 1, or 2.
 |  | 0 | 1 | 2 |  Total number of instances |
 | Train | 42,831 | 42,619 | 170 | 85,620 |
 | Test  | 10,681 | 10,680 | 44 | 21,405 |
 ## Data generation
+The Sage notebook within this [directory](https://github.com/pnnl/ML4AlgComb/tree/master/schubert_polynomial_structure)
+gives the code used to generate these datasets.
 The process involves:
 - For a chosen \\(n\\), compute the products \\(\mathfrak{S}_{\alpha} \mathfrak{S}_{\beta}\\) for \\(\alpha,\beta \in S_n\\).
+- For each of these pairs, extract and add to the dataset all non-zero structure constants \\(c^{\gamma_1}_{\alpha,\beta}, \dots, c^{\gamma_k}_{\alpha,\beta}\\).
 - Furthermore, for each \\(c^{\gamma_i}_{\alpha,\beta} \neq 0\\), randomly permute \\(\gamma_i \mapsto \gamma_i'\\) to find \\(c^{\gamma_i'}_{\alpha,\beta} = 0\\) and \\(c^{\gamma_i'}_{\alpha,\beta}\\) is not already in the dataset.
 ## Task
 **Math question:** Find a combinatorial interpretation of the structure constants \\(c_{\alpha,\beta}^\gamma\\)
+based on properties of \\(\alpha\\), \\(\beta\\), and \\(\gamma\\).
 **Narrow ML task:** Train a model that, given three permutations \\(\alpha, \beta, \gamma\\), can
+predict the associated structure constant \\(c^{\gamma}_{\alpha,\beta}\\). Extract the rules
+the model uses to make successful predictions.
 ## Small model performance
+Model and training details can be found in our paper.
 | Size | Logistic regression | MLP | Transformer | Guessing majority class |
 |----------|----------|-----------|------------|------------|