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Positivity of Pre-Canonical Bases for Spherical Hecke Algebras — arXiv:2608.07703

Source: arXiv (en)
Original: https://arxiv.org/abs/2608.07703
License (as archived): CC0-1.0
Archived: 2026-10-19T09:08:47.418Z
SHA-256 of the archived text: fcb38f9f228245437697886cad7dcf9e3378c31e6c25ecaa8df03702c2b151e7
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How to cite

Positivity of Pre-Canonical Bases for Spherical Hecke Algebras — arXiv:2608.07703 — arXiv (en), https://arxiv.org/abs/2608.07703 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_271f65cbfc79 (Metraipolis archive dc_271f65cbfc79, sha256 fcb38f9f228245437697886cad7dcf9e3378c31e6c25ecaa8df03702c2b151e7)

BibTeX

@misc{dc271f65cbfc79,
  title = {Positivity of Pre-Canonical Bases for Spherical Hecke Algebras — arXiv:2608.07703},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2608.07703},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_271f65cbfc79; sha256 fcb38f9f228245437697886cad7dcf9e3378c31e6c25ecaa8df03702c2b151e7},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2608.07703. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-08-11 in math.CO).
Categorie arXiv: Combinatorics (math.CO) ; Representation Theory (math.RT).
Pagina del lavoro: https://arxiv.org/abs/2608.07703 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-19T09:05:33.538Z dalla pagina https://arxiv.org/catchup/math.CO/2026-08-11?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).