ACDRepo/coefficients_of_kl_polynomials_6

The Coefficients of Kazhdan-Lusztig Polynomials for Permutations of Size 6

Kazhdan-Lusztig (KL) polynomials are polynomials in a variable $qq$ 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 $\sigma \backslash sigma$ and $\nu \backslash nu$ as $P_{\sigma ,\nu }(q)P\_\{\backslash sigma,\backslash nu\}(q)$. For example, the KL polynomial associated with permutations $\sigma =1432765109811\backslash sigma\; =\; 1\; \backslash ;\; 4\; \backslash ;\; 3\; \backslash ;\; 2\; \backslash ;\; 7\; \backslash ;\; 6\; \backslash ;\; 5\; \backslash ;\; 10\; \backslash ;\; 9\; \backslash ;\; 8\; \backslash ;\; 11$ and $\nu =4678910111235\backslash nu\; =\; 4\; \backslash ;\; 6\; \backslash ;\; 7\; \backslash ;\; 8\; \backslash ;\; 9\; \backslash ;\; 10\; \backslash ;\; 1\; \backslash ;\; 11\; \backslash ;\; 2\; \backslash ;\; 3\; \backslash ;\; 5$ is

$P_{\sigma ,\nu }(q)=1+16q+103q^2+337q^3+566q^4+529q^5+275q^6+66q^7+3q^{8}P\_\{\backslash sigma,\backslash nu\}(q)\; =\; 1\; +\; 16q\; +\; 103q^2\; +\; 337q^3\; +\; 566q^4\; +\; 529q^5\; +\; 275q^6\; +\; 66q^7\; +\; 3q^8$

(see here 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], nevertheless, in many ways they remain mysterious. For instance, there is no known closed formula for the degree of $P_{\sigma ,\nu }(q)P\_\{\backslash sigma,\backslash nu\}(q)$.

One family of questions revolve around the coefficients of $P_{\sigma ,\nu }(q)P\_\{\backslash sigma,\backslash nu\}(q)$. For instance, it has been hypothesized that the coefficient on the largest possible monomial term $q^{(\mathrm{\ell }(\sigma )-\mathrm{\ell }(\nu )-1)\mathrm{/}2}q^\{(\backslash ell(\backslash sigma)\; -\; \backslash ell(\backslash nu)-1)/2\}$ (where $\mathrm{\ell }(x)\backslash ell(x)$ is a statistic of the permutation $xx$ called the length of the permutation), which is known as the $\mu \backslash mu$-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 $n,x\in S_{n}n,x\; \backslash in\; S\_n$ along with the coefficients of the polynomial $P_{x,w}(q)P\_\{x,w\}(q)$. If $x=123456x\; =\; \backslash ;1\; \backslash ;2\; \backslash ;3\backslash ;\; 4\backslash ;\; 5\backslash ;\; 6$, $w=456123w=4\; \backslash ;5\backslash ;\; 6\backslash ;\; 1\; \backslash ;2\; \backslash ;3$ and $P_{v,w}(q)=1+4q+4q^2+q^{3}P\_\{v,w\}(q)\; =\; 1\; +\; 4q\; +\; 4q^2\; +\; q^3$ then the coefficients field is written as 1, 4, 4, 1. Note that coefficients are listed by increasing degree of the power of $qq$ (e.g., the coefficient on $11$ comes first, then the coefficient on $qq$, then the coefficient on $q^{2}q^2$, etc.)

We summarize the limited number of values coefficients on $P_{x,w}(q)P\_\{x,w\}(q)$ take when $x,w\in S_{6}x,\; w\; \backslash in\; S\_6$.

Constant Terms:

	0	1	Total number of instances
Train	336,071	78,649	414,720
Test	83,922	19,758	103,680

Coefficients on $qq$:

	0	1	2	3	4	Total number of instances
Train	397,386	13,253	3,483	535	63	414,720
Test	99,354	3,311	883	117	15	103,680

Coefficient on $q^{2}q^2$:

	0	1	2	3	4	Total number of instances
Train	412,707	1,705	242	40	26	414,720
Test	103,177	441	46	8	8	103,680

Coefficient on $q^{3}q^3$:

	0	1	Total number of instances
Train	414,688	32	414,720
Test	103,670	10	103,680

Data Generation

Datasets were generated using C code from Greg Warrington's website. The code we used can be found here.

Task

Math question: Generate conjectures around the properties of coefficients appearing on KL polynomials.

Narrow ML task: Predict the coefficients of $P_{x,w}(q)P\_\{x,w\}(q)$ given $xx$ and $ww$. 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 $xx$, $ww$, and the coefficients of $P_{x,w}(q)P\_\{x,w\}(q)$ has the potential to yield critical insights.

Small model performance

Since there are many possible tasks here, we did not run exhaustive hyperparameter searches. Instead, we ran ReLU MLPs with depth 4, width 256, and learning rate 0.0005.

Accuracy predicting coefficients for permutations of 6 elements:

Coefficient	MLP	Transformer	Guessing largest class
$11$	$99.9\mathrm{\%}\pm 0.0\mathrm{\%}99.9\backslash \%\; \backslash pm\; 0.0\backslash \%$	$100.0\mathrm{\%}\pm 0.0\mathrm{\%}100.0\backslash \%\; \backslash pm\; 0.0\backslash \%$	$80.9\mathrm{\%}80.9\backslash \%$
$qq$	$99.9\mathrm{\%}\pm 0.0\mathrm{\%}99.9\backslash \%\; \backslash pm\; 0.0\backslash \%$	$99.9\mathrm{\%}\pm 0.0\mathrm{\%}99.9\backslash \%\; \backslash pm\; 0.0\backslash \%$	$95.8\mathrm{\%}95.8\backslash \%$
$q^{2}q^2$	$99.9\mathrm{\%}\pm 0.0\mathrm{\%}99.9\backslash \%\; \backslash pm\; 0.0\backslash \%$	$99.9\mathrm{\%}\pm 0.0\mathrm{\%}99.9\backslash \%\; \backslash pm\; 0.0\backslash \%$	$99.5\mathrm{\%}99.5\backslash \%$
$q^{3}q^3$	$99.9\mathrm{\%}\pm 0.0\mathrm{\%}99.9\backslash \%\; \backslash pm\; 0.0\backslash \%$	$99.9\mathrm{\%}\pm 0.0\mathrm{\%}99.9\backslash \%\; \backslash pm\; 0.0\backslash \%$	$99.9\mathrm{\%}99.9\backslash \%$

The associated macro F1-scores are:

Coefficient	MLP	Transformer
$11$	$99.9\mathrm{\%}\pm 0.0\mathrm{\%}99.9\backslash \%\; \backslash pm\; 0.0\backslash \%$	$100.0\mathrm{\%}\pm 0.0\mathrm{\%}100.0\backslash \%\; \backslash pm\; 0.0\backslash \%$
$qq$	$99.0\mathrm{\%}\pm 1.5\mathrm{\%}99.0\backslash \%\; \backslash pm\; 1.5\backslash \%$	$98.0\mathrm{\%}\pm 3.7\mathrm{\%}98.0\backslash \%\; \backslash pm\; 3.7\backslash \%$
$q^{2}q^2$	$97.4\mathrm{\%}\pm 5.2\mathrm{\%}97.4\backslash \%\; \backslash pm\; 5.2\backslash \%$	$98.0\mathrm{\%}\pm 3.7\mathrm{\%}98.0\backslash \%\; \backslash pm\; 3.7\backslash \%$
$q^{3}q^3$	$87.9\mathrm{\%}\pm 4.5\mathrm{\%}87.9\backslash \%\; \backslash pm\; 4.5\backslash \%$	$88.3\mathrm{\%}\pm 17.1\mathrm{\%}88.3\backslash \%\; \backslash pm\; 17.1\backslash \%$

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

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, [email protected]

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.