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arXiv · 2608.07703

Positivity of Pre-Canonical Bases for Spherical Hecke Algebras

David Plaza·Yamil Sagurie
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Abstract

We introduce a new algorithm for computing Kostka-Foulkes polynomials. It arises from a combinatorial procedure that computes the transition coefficients between successive pre-canonical bases for spherical Hecke algebras, introduced by Libedinsky, Patimo, and the first author, in order to interpolate between the standard and canonical bases of the spherical Hecke algebras. Using this algorithm, we prove that the coefficients of the polynomials in the transition matrix from any pre-canonical basis to the next one has nonnegative coefficients. This establishes the positivity conjecture for pre-canonical bases and, moreover, yields a strictly stronger result that uncovers additional positivity phenomena beyond the original scope of the conjecture. As a by-product, this method yields a combinatorial algorithm for computing Lascoux's atomic decomposition of canonical basis elements. Finally, our results show that, after applying the Satake isomorphism and dualizing with respect to the Hall inner product, pre-canonical bases correspond to a new family of Schur-positive symmetric functions.

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David Plaza, Yamil Sagurie. 2026-08-07. Positivity of Pre-Canonical Bases for Spherical Hecke Algebras. https://arxiv.org/abs/2608.07703

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