Conjectures About Certain Parabolic Kazhdan–Lusztig Polynomials

Abstract

Irreducibility results for parabolic induction of representations of the general linear group over a local non-Archimedean field can be formulated in terms of Kazhdan–Lusztig polynomials of type A. Spurred by these results and some computer calculations, we conjecture that certain alternating sums of Kazhdan–Lusztig polynomials known as parabolic Kazhdan–Lusztig polynomials satisfy properties analogous to those of the ordinary ones.

Appendix: Numerical Results

We have calculated all the polynomials $\tilde P_{x,w}^{(m)}$, x, w ∈ S _n and verified Conjecture 1.1 for nm ≤ 12. ³(Recall that Conjecture 1.1 is known for n = 2.) Let us call a pair (w, x) in S _n reduced if it admits no cancelable indices (see Remark 1.2(1.2)) and xs < x (resp., sx < x) for any simple reflection s such that ws < w (resp., sw < w). In Tables 1, 2, 3, and 4, we list $\tilde P_{x,w}^{(m)}$ in the cases nm ≤ 12 (n, m > 1) for all reduced pairs (w, x) in S _n. By Conjecture 1.1 (which we checked at the cases at hand) and Remark 1.2(1.2), this covers all the polynomials $\tilde P_{x,w}^{(m)}$ without restriction on (w, x). To avoid repetitions, we only list representatives for the equivalence classes of the relation (w, x) ∼ (w ⁻¹, x ⁻¹) ∼ (w ₀ ww ₀, w ₀ xw ₀) ∼ (w ₀ w ⁻¹ w ₀, w ₀ x ⁻¹ w ₀).

Table 1 n = 4

Table 2 n = 5

Table 3 Cases for n = 6 where $\tilde P_{x,w}^{(2)}=(P_{x,w}^2+P_{x,w}(q^2))/2$

Table 4 Cases for n = 6 where $\tilde P_{x,w}^{(2)}\ne (P_{x,w}^2+P_{x,w}(q^2))/2$

Note that in the cases n = 4, 5 we have $\tilde P_{x,w}^{(2)}=(P_{x,w}^2+P_{x,w}(q^2))/2$. We split the case n = 6 according to two subcases.

1.1 Implementation

For the computation, we actually wrote and executed a C program to calculate all ordinary Kazhdan–Lusztig polynomials P _x,w for the symmetric groups S _k, k ≤ 12. As far as we know, this computation is already new for k = 11. (See [dC02] and [War11] for accounts of earlier computations, as well as the documentation of the Atlas software and other mathematical software packages.) As always, the computation proceeds with the original recursive formula of Kazhdan–Lusztig [KL79]

$$\displaystyle \begin{aligned} P_{x,w}=q^cP_{x,ws}+q^{1-c}P_{xs,ws}-\sum_{z:zs<z<ws}\mu(z,ws)q^{\frac{\ell(w)-\ell(z)}2}P_{x,z} \end{aligned}$$

where μ(x, y) is the coefficient of q ^{(ℓ(y)−ℓ(x)−1)∕2} (the largest possible degree) in P _x,y, s is a simple reflection such that ws < w and c is 1 if xs < x and 0 otherwise. However, some special features of the symmetric group allow for a faster, if ad hoc, code. (See Remark 1.2(1.2) and the comments below.) For S ₁₁, the program runs on a standard laptop (a Lenovo T470s 2017 model with 2.7 GHz CPU and 16GB RAM) in a little less than 3 h. For S ₁₂, the memory requirements are about 500 GB RAM. We ran it on the computer of the Faculty of Mathematics and Computer Science of the Weizmann Institute of Science (SGI, model UV-10), which has one terabyte RAM and 2.67 GHz CPU. The job was completed after almost a month of CPU time on a single core.

Let us give a few more details about the implementation. We say that a pair (w, x) is fully reduced if it is reduced (see above) and x ≤ ws, sw for any simple reflection s. Recall that we only need to compute P _x,w for fully reduced pairs. The number of fully reduced pairs for S ₁₂, up to symmetry, is about 46 × 10⁹. However, a posteriori, the number of distinct polynomials obtained is “only” about 4.3 × 10⁹. This phenomenon (which had been previously observed for smaller symmetric groups) is crucial for the implementation since it makes the memory requirements feasible. An equally important feature, which once again had been noticed before for smaller symmetric groups, is that only for a small fraction of the pairs above, namely about 66.5 × 10⁶, we have μ(x, w) > 0. This fact cuts down significantly the number of summands in the recursive formula and makes the computation feasible in terms of time complexity.

We store the results as follows:

• 1. A “glossary” of the ∼ 4.3 × 10⁹ different polynomials. (The coefficients of the vast majority of the polynomials are smaller than 2¹⁶ = 65536. The average degree is about 10.)

• 2. A table with ∼ 46 × 10⁹ entries that provides for each reduced pair the pointer to P _x,w in the glossary.

• 3. An additional lookup table of size 12! ∼ 0.5 × 10⁹ (which is negligible compared to the previous one) so that in the previous table we only need to record x and the pointer to P _x,w (which can be encoded in 29 and 33 bits, respectively), but not w.

• 4. A table with ∼ 66.5 × 10⁶ entries recording x, w, μ(x, w) for all fully reduced pairs (up to symmetry) with μ(x, w) > 0.

Thus, the main table is of size ∼ 8 × 46 × 10⁹ bytes, or about 340 GB. This is supplemented by the glossary table which is of size < 100 GB, plus auxiliary tables of insignificant size. Of course, by the nature of the recursive algorithm all these tables have to be stored in the RAM.

We mention a few additional technical aspects about the program.

• 1. The outer loop is over all permutation w ∈ S _n in lexicographic order. Given w ∈ S ₁₂, it is possible to enumerate efficiently the pairs (w, x) such that xs < x (resp., sx < x) whenever ws < w (resp., sw < w). More precisely, given such x < w we can very quickly find the next such x in lexicographic order. Moreover, one can incorporate the “non-cancelability” condition to this “advancing” procedure and then test the condition x ≤ ws, sw for the remaining x’s. Thus, it is perfectly feasible to enumerate the ∼ 46 × 10⁹ fully reduced pairs.

• 2. On the surface, the recursive formula requires a large number of additions and multiplications in each step. However, in reality, the number of summands is usually relatively small, since the μ-function is rarely nonzero.

• 3. For each w≠1, the program picks the first simple root s (in the standard ordering) such that ws < w and produces the list of z’s such that zs < z < ws and μ(z, ws) > 0. The maximal size of this list turns out to be ∼ 100, 000 but it is usually much much smaller. The list is then used to compute P _x,w (and in particular, μ(x, w)) for all fully reduced pairs using the recursion formula and the polynomials already generated for w ^′ < w. Of course, for any given x only the z’s with x ≤ z matter.

• 4. Since we only keep the data for fully reduced pairs (in order to save memory), we need to find, for any given pair, the fully reduced pair which “represents” it. Fortunately, this procedure is reasonably quick.

• 5. The glossary table is continuously updated and stored as 1000 binary search trees, eventually consisting of ∼ 4.3 × 10⁶ internal nodes each. The data is sufficiently random so that there is no need to balance the trees. The memory overhead for maintaining the trees is inconsequential.

• 6. In principle, it should be possible to parallelize the program so that it runs simultaneously on many processors. The point is that the recursive formula only requires the knowledge of $P_{x^{\prime },w^{\prime }}$ with w ^′ < w, so we can compute all P _x,w’s with a fixed ℓ(w) in parallel. For technical reasons, we have not been able to implement this parallelization.

As a curious by-product of our computation, we get (Table 5).

Table 5 Values of μ(x, w) and pairs attaining them for S ₁₂

Corollary 8.1

The values of μ(x, w) for x, w ∈ S ₁₂ are given by

$$\displaystyle \begin{aligned} \{1,2,3,4,5,6,7,8,9,10,17,18,19,20,21,22,23,24,25,26,27,28,158,163\}. \end{aligned}$$

This complements [War11, Theorem 1.1]. The new values of μ are 9, 10, 17, 19, 20, 21, 22.

Complete tables listing the fully reduced pairs in S _k, k ≤ 12 with μ > 0 (together with their μ-value) are available upon request. The size of the compressed file for S ₁₂ is 200MB.

Finally, I would like to take this opportunity to thank Amir Gonen, the Unix system engineer of our faculty, for his technical assistance with running this heavy-duty job.