| --- |
| license: cc-by-2.0 |
| pretty_name: Coefficients on Kazhdan–Lusztig polynomials for permutations of size 7 |
| --- |
| |
| # The Coefficients of Kazhdan-Lusztig Polynomials for Permutations of Size 7 |
|
|
| Kazhdan-Lusztig (KL) polynomials are polynomials in a variable \\(q\\) and |
| with integer coefficients that (for our purposes) are indexed by a pair of permutations [1]. |
| We will write the KL polynomial associated with permutations \\(\sigma\\) and \\(\nu\\) as |
| \\(P_{\sigma,\nu}(q)\\). For example, the KL polynomial associated with permutations |
| \\(\sigma = 1 \; 4 \; 3 \; 2 \; 7 \; 6 \; 5 \; 10 \; 9 \; 8 \; 11\\) and |
| \\(\nu = 4 \; 6 \; 7 \; 8 \; 9 \; 10 \; 1 \; 11 \; 2 \; 3 \; 5\\) is |
| |
| |
| \\(P_{\sigma,\nu}(q) = 1 + 16q + 103q^2 + 337q^3 + 566q^4 + 529q^5 + 275q^6 + 66q^7 + 3q^8\\) |
|
|
| (see [here](https://gswarrin.w3.uvm.edu/research/klc/klc.html) for efficient software to compute |
| these polynomials). KL polynomials have deep connections throughout several areas of mathematics. For example, |
| KL polynomials are related to the dimensions of intersection homology in Schubert calculus, |
| the study of the Hecke algebra, and representation theory of the symmetric group. They |
| can be computed via a recursive formula [[1]](https://link.springer.com/article/10.1007/BF01390031), |
| nevertheless, in many ways they remain mysterious. For instance, there is no known closed |
| formula for the degree of \\(P_{\sigma,\nu}(q)\\). |
| |
| One family of questions revolve around the coefficients of \\(P_{\sigma,\nu}(q)\\). |
| For instance, it has been hypothesized that the coefficient on the largest possible monomial term |
| \\(q^{(\ell(\sigma) - \ell(\nu)-1)/2}\\) (where \\(\ell(x)\\) is a statistic of the |
| permutation \\(x\\) called the *length* of the permutation), which is known as the |
| \\(\mu\\)-coefficient, has a combinatorial interpretation but currently this is not |
| known. Better understanding this and other coefficients is of significant |
| interest to mathematicians from a range of fields. |
|
|
| ## Dataset details |
|
|
| Each instance in this dataset consists of a pair of permutations of \\(n,x \in S_n\\) |
| along with the coefficients of the polynomial \\(P_{x,w}(q)\\). If \\(x = \;1 \;2 \;3\; 4\; 5\; 6\\), |
| \\(w=4 \;5\; 6\; 1 \;2 \;3\\) and \\(P_{v,w}(q) = 1 + 4q + 4q^2 + q^3\\) |
| then the coefficients field is written as `1, 4, 4, 1`. Note that coefficients are listed |
| by increasing degree of the power of \\(q\\) (e.g., the coefficient on \\(1\\) comes first, |
| then the coefficient on \\(q\\), then the coefficient on \\(q^2\\), etc.) |
| |
| We summarize the limited number of values coefficients on \\(P_{x,w}(q)\\) take when \\(x, w \in S_7\\). |
| |
| **Constant Terms:** |
| |
| | | 0 | 1 | Total number of instances | |
| |----------|----------|----------|----------| |
| | Train | 17,479,910 | 2,841,370 | 20,321,280 | |
| | Test | 4,370,771 | 709,549 | 5,080,320 | |
| |
| **Coefficients on \\(q\\):** |
| |
| | | 0 | 1 | 2 | 3 | 4 | 5 | 6 | Total number of instances | |
| |----------|----------|----------|----------|----------|----------|----------|----------|----------| |
| | Train | 19,291,150 | 660,600 | 266,591 | 80,173 | 18,834 | 3,221 | 711 | 20,321,280 | |
| | Test | 4,822,214 | 165,768 | 66,593 | 19,963 | 4,762 | 819 | 201 | 5,080,320 | |
| |
| **Coefficient on \\(q^2\\):** |
| |
| | | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | Total number of instances | |
| |----------|----------|----------|----------|----------|----------|----------|----------|----------|----------|----------|----------|----------|----------|----------|----------|----------| |
| | Train | 20,072,738 | 170,412 | 46,226 | 16,227 | 7,621 | 4,023 | 1,287 | 1,153 | 785 | 350 | 152 | 139 | 121 | 42 | 4 | 20,321,280 | |
| | Test | 5,017,962 | 42,748 | 11,568 | 4,021 | 1,905 | 1,065 | 349 | 287 | 183 | 86 | 40 | 37 | 47 | 22 | 5,080,320 | |
| |
| **Coefficient on \\(q^3\\):** |
| |
| | | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 15 | Total number of instances | |
| |----------|----------|----------|----------|----------|----------|----------|----------|----------|----------|----------|----------|----------|----------|----------| |
| | Train | 20,291,535 | 22,094 | 4,779 | 1,660 | 590 | 195 | 206 | 115 | 34 | 26 | 24 | 18 | 4 | 20,321,280 | |
| | Test | 507,2831 | 5,498 | 1,213 | 442 | 146 | 61 | 50 | 37 | 14 | 6 | 8 | 14 | 5,080,320 | |
| |
| **Coefficient on \\(q^4\\):** |
| |
| | | 0 | 1 | Total number of instances | |
| |----------|----------|----------|----------| |
| | Train | 17,479,910 | 2,841,370 | 20,321,280 | |
| | Test | 4,370,771 | 709,549 | 5,080,320 | |
| |
| |
| ## Data Generation |
| |
| Datasets were generated using C code from Greg Warrington's |
| [website](https://gswarrin.w3.uvm.edu/research/klc/klc.html). The code we used can be found |
| [here](https://github.com/pnnl/ML4AlgComb/tree/master/kl-polynomial_coefficients). |
| |
| ## Task |
| |
| **Math question:** Generate conjectures around the properties of coefficients appearing on KL polynomials. |
| |
| **Narrow ML task:** Predict the coefficients of \\(P_{x,w}(q)\\) given \\(x\\) and \\(w\\). |
| We break this up into a separate task for each coefficient though one could |
| choose to predict all simultaneously. Since there are generally very few |
| different integers that arise as coefficients (at least in these small examples), |
| we frame this problem as one of classification. |
|
|
| While the classification task as framed does not capture the broader math question exactly, |
| illuminating connections between \\(x\\), \\(w\\), and the coefficients of \\(P_{x,w}(q)\\) |
| has the potential to yield critical insights. |
| |
| ## Further information |
| |
| - **Curated by:** Henry Kvinge |
| - **Funded by:** Pacific Northwest National Laboratory |
| - **Language(s) (NLP):** NA |
| - **License:** CC-by-2.0 |
| |
| ## 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 |
| |
| \[1\] Kazhdan, David, and George Lusztig. "Representations of Coxeter groups and Hecke algebras." Inventiones mathematicae 53.2 (1979): 165-184. |
| [2] Warrington, Gregory S. "Equivalence classes for the μ-coefficient of Kazhdan–Lusztig polynomials in Sn." Experimental Mathematics 20.4 (2011): 457-466. |
