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Schubert puzzles and integrability III: separated descents

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arxiv 2306.13855 v1 pith:37DOWSZ3 submitted 2023-06-24 math.CO

Allen Knutson , Paul Zinn-Justin This is my paper · ORCID
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keywords descentformulaeseparatedschubertbeforecasedescentsfirst
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abstract

In paper I of this series we gave positive formulae for expanding the product $\mathfrak S^\pi \mathfrak S^\rho$ of two Schubert polynomials, in the case that both $\pi,\rho$ had shared descent set of size $\leq 3$. Here we introduce and give positive formulae for two new classes of Schubert product problems: separated descent in which $\pi$'s last descent occurs at (or before) $\rho$'s first, and almost separated descent in which $\pi$'s last two descents occur at (or before) $\rho$'s first two respectively. In both cases our puzzle formulae extend to $K$-theory (multiplying Grothendieck polynomials), and in the separated descent case, to equivariant $K$-theory. The two formulae arise (via quantum integrability) from fusion of minuscule quantized loop algebra representations in types $A$, $D$ respectively.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Graham positivity of triple Schubert calculus

    math.CO 2025-06 conditional novelty 8.0 of 10

    The paper proves Samuel's conjecture that triple Schubert calculus coefficients lie in the semiring N[t_i - y_j], and derives Kirillov's conjecture on the positivity of skew divided difference operators.

  2. Equivariant Schubert Calculus for Inverse Grassmannian Permutations

    math.CO 2026-07 conditional novelty 7.0 of 10

    An equivariant product rule: double Schubert polynomials indexed by inverse Grassmannian permutations expand with structure constants given by double Schubert polynomials in two disjoint sets of variables.

  3. Richardson tableaux and Schubert positivity

    math.CO 2025-10 conditional novelty 7.0 of 10

    The Schubert cycle expansion of a Springer fiber component equal to a Richardson variety, indexed by a Richardson tableau, is computed combinatorially using translation-equivalent Bruhat intervals and Sottile's Pieri rule.

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