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README.md

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-license: cc-by-2.0
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+---
+license: cc-by-2.0
+pretty_name: structure constants of schubert polynomials, n = 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\\). 
+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}\\) and \\(\mathfrak{S}_{\beta}\\) 
+(indexed by permutations \\(\alpha \in S_n\\) and \\(\beta \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^{\gamma}_{\alpha \beta} \mathfrak{S}_{\gamma}\\). 
+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
+
+We multiply Schubert polynomials corresponding to permutations of \\(\{1,2,3\}\\), 
+\\(\alpha = 2 1 3\\) and \\(\beta = 1 3 2\\). Writing these in terms of indeterminants 
+\\(x_1\\), \\(x_2\\), and \\(x_3\\), we have \\(\mathfrak{S}_{\alpha} = x_1 + x_2\\) 
+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 
+Each instance in this dataset is a triple of permutations \\((\alpha,\beta,\gamma)\\), 
+labeled by its coefficient \\(c^{\gamma}_{\alpha \beta}\\) in the expansion of the product 
+\\(\mathfrak{S}_{\alpha} \mathfrak{S}_{\beta}\\). We call permutations \\(\alpha\\) and \\(\beta\\)
+*lower index permutations 1* and *2* respectively. We call \\(\gamma\\) the *upper index 
+permutation*. The datasets are organized so that 
+\\(\alpha\\) and \\(\beta\\) are always drawn from the symmetric group on \\(n\\) elements 
+(we provide datasets for \\(n = 3\\), \\(4\\), and \\(5\\)), but \\(\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 | 4,202,040 | 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 | 
+|----------|----------|-----------|------------|------------|
+| \\(n= 4\\) | \\(88.8\%\\) | \\(93.1\% \pm 2.6\%\\) | \\(94.6\% \pm 1.0\%\\) | \\(52.3\%\\) |
+| \\(n= 5\\) | \\(90.6\%\\) | \\(97.5\% \pm 0.2\%\\) | \\(96.2\% \pm 1.1\%\\) | \\(49.9\%\\) |
+| \\(n= 6\\) | \\(89.7\%\\) | \\(99.8\% \pm 0.0\%\\) | \\(91.3\% \pm 8.0\%\\) | \\(50.1\%\\) |
+
+The \\(\pm\\) signs indicate 95% confidence intervals from random weight initialization and training.
+
+- **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](https://github.com/pnnl/ML4AlgComb/tree/master/schubert_polynomial_structure).
+
+- **Repository:** [ACD Repo](https://github.com/pnnl/ML4AlgComb/tree/master)
+
+## 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.