ACDRepo/schubert_polynomial_structure_constants_6

A Combinatorial Interpretation of Schubert Polynomial Structure Constants

Schubert polynomials [1,2,3] are a family of polynomials indexed by permutations of $S_{n}S\_n$. Developed to study the cohomology ring of the flag variety, they have deep connections to algebraic geometry, Lie theory, and representation theory. Despite their geometric origins, Schubert polynomials can be described combinatorially [4,5], making them a well-studied object in algebraic combinatorics. An important open problem in the study of Schubert polynomials is understanding their structure constants.

When two Schubert polynomials ${\mathfrak{S}}_{\alpha }\backslash mathfrak\{S\}\_\{\backslash alpha\}$ and ${\mathfrak{S}}_{\beta }\backslash mathfrak\{S\}\_\{\backslash beta\}$ (indexed by permutations $\alpha \in S_n\backslash alpha\; \backslash in\; S\_n$ and $\beta \in S_m\backslash beta\; \backslash in\; S\_m$) are multiplied, their product can be written as a linear combination of Schubert polynomials ${\mathfrak{S}}_{\alpha }{\mathfrak{S}}_{\beta }={\sum }_{\gamma }{c}_{\alpha \beta }^{\gamma }{\mathfrak{S}}_{\gamma }\backslash mathfrak\{S\}\_\{\backslash alpha\}\; \backslash mathfrak\{S\}\_\{\backslash beta\}\; =\; \backslash sum\_\{\backslash gamma\}\; c^\{\backslash gamma\}\_\{\backslash alpha\; \backslash beta\}\; \backslash mathfrak\{S\}\_\{\backslash gamma\}$. where the sum runs over permutations in $S_{n+m}S\_\{n+m\}$. The question is whether the $c_{\alpha \beta }^{\gamma }c^\{\backslash gamma\}\_\{\backslash alpha\; \backslash beta\}$ (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).

Example

We multiply Schubert polynomials corresponding to permutations of $\{1,2,3\}\backslash \{1,2,3\backslash \}$, $\alpha =213\backslash alpha\; =\; 2\; 1\; 3$ and $\beta =132\backslash beta\; =\; 1\; 3\; 2$, each written in one line notation. Writing these in terms of indeterminants $x_{1}x\_1$, $x_{2}x\_2$, and $x_{3}x\_3$, we have ${\mathfrak{S}}_{\alpha }=x_1+x_2\backslash mathfrak\{S\}\_\{\backslash alpha\}\; =\; x\_1\; +\; x\_2$ and ${\mathfrak{S}}_{\beta }=x_1\backslash mathfrak\{S\}\_\{\backslash beta\}\; =\; x\_1$. Multiplying these together we get ${\mathfrak{S}}_{\alpha }{\mathfrak{S}}_{\beta }=x_1^2+x_1{x}_2\backslash mathfrak\{S\}\_\{\backslash alpha\}\backslash mathfrak\{S\}\_\{\backslash beta\}\; =\; x\_1^2\; +\; x\_1x\_2$. As ${\mathfrak{S}}_{231}=x_1{x}_2\backslash mathfrak\{S\}\_\{2\; 3\; 1\}\; =\; x\_1x\_2$ and ${\mathfrak{S}}_{312}=x_1^2\backslash mathfrak\{S\}\_\{3\; 1\; 2\}\; =\; x\_1^2$ we can write ${\mathfrak{S}}_{\alpha }{\mathfrak{S}}_{\beta }={\mathfrak{S}}_{231}+{\mathfrak{S}}_{312}\backslash mathfrak\{S\}\_\{\backslash alpha\}\backslash mathfrak\{S\}\_\{\backslash beta\}\; =\; \backslash mathfrak\{S\}\_\{2\; 3\; 1\}\; +\; \backslash mathfrak\{S\}\_\{3\; 1\; 2\}$. It follows that for these $\alpha \backslash alpha$ and $\beta \backslash beta$, $c_{\alpha ,\beta }^{\gamma }=1c\_\{\backslash alpha,\backslash beta\}^\{\backslash gamma\}\; =\; 1$ if $\gamma =231\backslash gamma\; =\; 2\; 3\; 1$ or $\gamma =312\backslash gamma\; =\; 3\; 1\; 2$ and otherwise $c_{\alpha ,\beta }^{\gamma }=0c\_\{\backslash alpha,\backslash beta\}^\{\backslash gamma\}\; =\; 0$.

Dataset

Each instance in this dataset is a triple of permutations $(\alpha ,\beta ,\gamma )(\backslash alpha,\backslash beta,\backslash gamma)$, labeled by its coefficient $c_{\alpha \beta }^{\gamma }c^\{\backslash gamma\}\_\{\backslash alpha\; \backslash beta\}$ in the expansion of the product ${\mathfrak{S}}_{\alpha }{\mathfrak{S}}_{\beta }\backslash mathfrak\{S\}\_\{\backslash alpha\}\; \backslash mathfrak\{S\}\_\{\backslash beta\}$. We call permutations $\alpha \backslash alpha$ and $\beta \backslash beta$ lower index permutations 1 and 2 respectively. We call $\gamma \backslash gamma$ the upper index permutation. The datasets are organized so that $\alpha \backslash alpha$ and $\beta \backslash beta$ are always drawn from the symmetric group on $nn$ elements, but $\gamma \backslash gamma$ may belong to a strictly larger symmetric group. Not all possible triples of permutations are included since the vast majority of these would be zero. The dataset consists of an approximately equal number of zero and nonzero coefficients (but they are not balanced between quantities of non-zero coefficients).

Statistics 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	4,198,767	4,092,744	108,818	2,290	9	3	8,402,631
Test	1,050,418	1,022,187	27,509	540	3	0	2,100,657

Data generation

The Sage notebook within this directory gives the code used to generate these datasets. The process involves:

• For a chosen $nn$, compute the products ${\mathfrak{S}}_{\alpha }{\mathfrak{S}}_{\beta }\backslash mathfrak\{S\}\_\{\backslash alpha\}\; \backslash mathfrak\{S\}\_\{\backslash beta\}$ for $\alpha ,\beta \in S_n\backslash alpha,\backslash beta\; \backslash in\; S\_n$.
• For each of these pairs, extract and add to the dataset all non-zero structure constants $c_{\alpha ,\beta }^{{\gamma }_1},\dots ,c_{\alpha ,\beta }^{{\gamma }_k}c^\{\backslash gamma\_1\}\_\{\backslash alpha,\backslash beta\},\; \backslash dots,\; c^\{\backslash gamma\_k\}\_\{\backslash alpha,\backslash beta\}$.
• Furthermore, for each $c_{\alpha ,\beta }^{{\gamma }_i}\mathrm{\ne }0c^\{\backslash gamma\_i\}\_\{\backslash alpha,\backslash beta\}\; \backslash neq\; 0$, randomly permute ${\gamma }_i\mapsto {\gamma }_i^{\mathrm{\prime }}\backslash gamma\_i\; \backslash mapsto\; \backslash gamma\_i\text{'}$ to find $c_{\alpha ,\beta }^{{\gamma }_i^{\mathrm{\prime }}}=0c^\{\backslash gamma\_i\text{'}\}\_\{\backslash alpha,\backslash beta\}\; =\; 0$ and $c_{\alpha ,\beta }^{{\gamma }_i^{\mathrm{\prime }}}c^\{\backslash gamma\_i\text{'}\}\_\{\backslash alpha,\backslash beta\}$ is not already in the dataset.

Task

Math question: Find a combinatorial interpretation of the structure constants $c_{\alpha ,\beta }^{\gamma }c\_\{\backslash alpha,\backslash beta\}^\backslash gamma$ based on properties of $\alpha \backslash alpha$, $\beta \backslash beta$, and $\gamma \backslash gamma$.
Narrow ML task: Train a model that, given three permutations $\alpha ,\beta ,\gamma \backslash alpha,\; \backslash beta,\; \backslash gamma$, can predict the associated structure constant $c_{\alpha ,\beta }^{\gamma }c^\{\backslash gamma\}\_\{\backslash alpha,\backslash beta\}$. 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
$n=4n=\; 4$	$88.8\mathrm{\%}88.8\backslash \%$	$93.1\mathrm{\%}\pm 2.6\mathrm{\%}93.1\backslash \%\; \backslash pm\; 2.6\backslash \%$	$94.6\mathrm{\%}\pm 1.0\mathrm{\%}94.6\backslash \%\; \backslash pm\; 1.0\backslash \%$	$52.3\mathrm{\%}52.3\backslash \%$
$n=5n=\; 5$	$90.6\mathrm{\%}90.6\backslash \%$	$97.5\mathrm{\%}\pm 0.2\mathrm{\%}97.5\backslash \%\; \backslash pm\; 0.2\backslash \%$	$96.2\mathrm{\%}\pm 1.1\mathrm{\%}96.2\backslash \%\; \backslash pm\; 1.1\backslash \%$	$49.9\mathrm{\%}49.9\backslash \%$
$n=6n=\; 6$	$89.7\mathrm{\%}89.7\backslash \%$	$99.8\mathrm{\%}\pm 0.0\mathrm{\%}99.8\backslash \%\; \backslash pm\; 0.0\backslash \%$	$91.3\mathrm{\%}\pm 8.0\mathrm{\%}91.3\backslash \%\; \backslash pm\; 8.0\backslash \%$	$50.1\mathrm{\%}50.1\backslash \%$

The $\pm \backslash pm$ signs indicate 95% confidence intervals from random weight initialization and training.

Further information

• Curated by: Henry Kvinge
• Funded by: Pacific Northwest National Laboratory
• Language(s) (NLP): NA
• License: CC-by-2.0

Dataset Sources

Data generation scripts can be found here.

• Repository: ACD Repo

Citation

BibTeX:

@article{chau2025machine,
    title={Machine learning meets algebraic combinatorics: A suite of datasets capturing research-level conjecturing ability in pure mathematics},
    author={Chau, Herman and Jenne, Helen and Brown, Davis and He, Jesse and Raugas, Mark and Billey, Sara and Kvinge, Henry},
    journal={arXiv preprint arXiv:2503.06366},
    year={2025}
}

APA:

Chau, H., Jenne, H., Brown, D., He, J., Raugas, M., Billey, S., & Kvinge, H. (2025). Machine learning meets algebraic combinatorics: A suite of datasets capturing research-level conjecturing ability in pure mathematics. arXiv preprint arXiv:2503.06366.

Dataset Card Contact

Henry Kvinge, acdbenchdataset@gmail.com

References

[1] Bernstein, IMGI N., Israel M. Gel'fand, and Sergei I. Gel'fand. "Schubert cells and cohomology of the spaces G/P." Russian Mathematical Surveys 28.3 (1973): 1.
[2] Demazure, Michel. "Désingularisation des variétés de Schubert généralisées." Annales scientifiques de l'École Normale Supérieure. Vol. 7. No. 1. 1974.
[3] Lascoux, Alain, and Marcel-Paul Schützenberger. "Structure de Hopf de l’anneau de cohomologie et de l’anneau de Grothendieck d’une variété de drapeaux." CR Acad. Sci. Paris Sér. I Math 295.11 (1982): 629-633.
[4] Billey, Sara C., William Jockusch, and Richard P. Stanley. "Some combinatorial properties of Schubert polynomials." Journal of Algebraic Combinatorics 2.4 (1993): 345-374.
[5] Bergeron, Nantel, and Sara Billey. "RC-graphs and Schubert polynomials." Experimental Mathematics 2.4 (1993): 257-269.