README.md

---
license: cc-by-2.0
pretty_name: Coefficients on Kazhdan–Lusztig polynomials for permutations of size 5
---

# The Coefficients of Kazhdan-Lusztig Polynomials for Permutations of Size 5

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 of coefficients on \\(P_{x,w}(q)\\) take when \\(x, w \in S_5\\). 

**Constant Terms:**

|  | 0 | 1 | Total number of instances | 
|----------|----------|----------|----------|
| Train | 8,496 | 3,024 | 11,520 |
| Test  | 2,123 | 757 | 2,880 |

**Coefficients on \\(q\\):**

|  | 0 | 1 | 2 | Total number of instances | 
|----------|----------|----------|----------|----------|
| Train | 11,219 | 267 | 34 | 11,520 |
| Test  | 2,793 | 77 | 10 | 2,880 |

**Coefficient on \\(q^2\\):**

|  | 0 | 1 | Total number of instances | 
|----------|----------|----------|----------|
| Train | 11,514 | 6 | 11,520 |
| Test  | 2,876 | 4 | 2,880 |

### Kazhdan-Lusztig Polynomials for Permutations of \\(6\\) elements

We summarize the limited number of values coefficients on \\(P_{x,w}(q)\\) take when \\(x, w \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 \\(q\\):**

|  | 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\\):**

|  | 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\\):**

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

### Kazhdan-Lusztig Polynomials for Permutations of \\(7\\) elements

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.

## 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.

### Kazhdan-Lusztig Polynomials for Permutations of \\(5\\) elements

Accuracy predicting coefficients for permutations of 5 elements:

| Coefficient | MLP | Transformer | Guessing largest class | 
|----------|----------|-----------|------------|
| \\(1\\) | \\(99.8\% \pm 0.2\%\\) | \\(99.9\% \pm 0.1\%\\) | \\(73.7\%\\) | 
| \\(q\\) | \\(99.5\% \pm 0.4\%\\) | \\(99.2\% \pm 1.0\%\\) | \\(97.0\%\\) |
| \\(q^2\\) | \\(99.9\% \pm 0.1\%\\) | \\(100.0\% \pm 0.0\%\\) | \\(99.9\%\\) |

The associated macro F1-scores are:

| Coefficient | MLP | Transformer | 
|----------|----------|-----------|
| \\(1\\) | \\(99.7\% \pm 0.1\%\\) | \\(99.9\% \pm 0.4\%\\) | 
| \\(q\\) | \\(93.9\% \pm 3.7\%\\) | \\(92.7\% \pm 7.6\%\\) | 
| \\(q^2\\) | \\(50.0\% \pm 0.0\%\\) | \\(100.0\% \pm 0.0\%\\) | 

### Kazhdan-Lusztig Polynomials for Permutations of \\(6\\) elements

Accuracy predicting coefficients for permutations of 6 elements:

| Coefficient | MLP | Transformer | Guessing largest class | 
|----------|----------|-----------|------------|
| \\(1\\) | \\(99.9\% \pm 0.0\%\\) | \\(100.0\% \pm 0.0\%\\) | \\(80.9\%\\)| 
| \\(q\\) | \\(99.9\% \pm 0.0\%\\) | \\(99.9\% \pm 0.0\%\\) | \\(95.8\%\\) |
| \\(q^2\\) | \\(99.9\% \pm 0.0\%\\) | \\(99.9\% \pm 0.0\%\\) | \\(99.5\%\\) |
| \\(q^3\\) | \\(99.9\% \pm 0.0\%\\) | \\(99.9\% \pm 0.0\%\\) | \\(99.9\%\\) |

The associated macro F1-scores are:

| Coefficient | MLP | Transformer | 
|----------|----------|-----------|
| \\(1\\) | \\(99.9\% \pm 0.0\%\\) | \\(100.0\% \pm 0.0\%\\) | 
| \\(q\\) | \\(99.0\% \pm 1.5\%\\) | \\(98.0\% \pm 3.7\%\\) | 
| \\(q^2\\) | \\(97.4\% \pm 5.2\%\\) | \\(98.0\% \pm 3.7\%\\) | 
| \\(q^3\\) | \\(87.9\% \pm 4.5\%\\) | \\(88.3\% \pm 17.1\%\\) | 

The \\(\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, 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.