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Richardson tableaux and Schubert positivity

Hunter Spink, Vasu Tewari

TL;DR

This work computes the Schubert cycle expansions of irreducible Springer fiber components that are Richardson varieties, generalizing Güemes’ hook-shape result and addressing a question of Karp–Precup. It introduces well-aligned pairs $(v,w)$ to organize Bruhat intervals, proves translation-equivalence to dominant cases, and provides two nonnegative combinatorial routes to compute $c^w_{u,v}$: (i) a Pieri-based translation-equivalence approach; (ii) Bergeron–Sottile maps realized via geometric push-pull and pattern maps. The authors then connect these combinatorics to Richardson tableaux by showing every Richardson tableau yields a well-aligned pair $(v_T,w_T)$, enabling explicit expansions of $[X^w_v]$ in the Schubert basis. They extend smoothness results to the broader class of very well-aligned pairs and describe geometric constructions that mirror the combinatorial procedures, linking Schubert calculus, Springer fiber geometry, and pattern-map techniques. Overall, the paper provides explicit, nonnegative, combinatorial rules for a broad family of Schubert coefficients associated with Richardson varieties, with significant implications for the interplay between geometry and combinatorics in type A flag varieties.

Abstract

We compute the Schubert cycle expansion of those irreducible components of Springer fibers equal to Richardson varieties. This generalizes work of Güemes in the case of a hook shape and answers a question of Karp-Precup.

Richardson tableaux and Schubert positivity

TL;DR

This work computes the Schubert cycle expansions of irreducible Springer fiber components that are Richardson varieties, generalizing Güemes’ hook-shape result and addressing a question of Karp–Precup. It introduces well-aligned pairs to organize Bruhat intervals, proves translation-equivalence to dominant cases, and provides two nonnegative combinatorial routes to compute : (i) a Pieri-based translation-equivalence approach; (ii) Bergeron–Sottile maps realized via geometric push-pull and pattern maps. The authors then connect these combinatorics to Richardson tableaux by showing every Richardson tableau yields a well-aligned pair , enabling explicit expansions of in the Schubert basis. They extend smoothness results to the broader class of very well-aligned pairs and describe geometric constructions that mirror the combinatorial procedures, linking Schubert calculus, Springer fiber geometry, and pattern-map techniques. Overall, the paper provides explicit, nonnegative, combinatorial rules for a broad family of Schubert coefficients associated with Richardson varieties, with significant implications for the interplay between geometry and combinatorics in type A flag varieties.

Abstract

We compute the Schubert cycle expansion of those irreducible components of Springer fibers equal to Richardson varieties. This generalizes work of Güemes in the case of a hook shape and answers a question of Karp-Precup.
Paper Structure (17 sections, 26 theorems, 61 equations, 5 figures)

This paper contains 17 sections, 26 theorems, 61 equations, 5 figures.

Table of Contents

  1. Combinatorial background
  2. Young tableaux
  3. Permutations
  4. Rothe diagrams and dominant permutations
  5. Schubert polynomials
  6. Well-aligned pairs
  7. Translation-equivalent Bruhat intervals and well-aligned pairs
  8. Labeled plane binary trees and dominant permutations
  9. Well-aligned Schubert structure coefficients
  10. Coefficients via translation-equivalence
  11. Coefficients via Bergeron--Sottile maps
  12. Richardson tableaux and well-aligned pairs
  13. The geometry of well and very-well aligned Richardson varieties
  14. Translation equivalence for well--aligned Richardson varieties
  15. Building well-aligned Richardsons from pattern maps and geometric push-pull
  16. ...and 2 more sections

Key Result

Theorem 1

Given a Richardson tableau $T$, there is a combinatorial rule for computing $c^{w_T}_{u,v_T}$.

Figures (5)

  • Figure 1: The Rothe diagrams for $43152$ (left) and dominant $43125$ (right).
  • Figure 2: The gray shaded cells correspond to the Rothe diagram of the dominant permutations $v=4521367$ and $\delta(v)=341256$ respectively, while the union of the gray and blue shaded cells correspond to the Rothe diagrams of $w=5724316$ and $\delta(w)=461325$ respectively.
  • Figure 3: A plane binary tree $T$ and the set $Z_T$.
  • Figure 4: Three decreasing trees with the leftmost tree associated to $w=2143$ and the rightmost corresponding to the dominant permutation $w^{\uparrow}=s_1s_2w=3241$.
  • Figure 5: The product $x_1^3x_2^3\mathfrak{S}_{1476235}$ computed via Corollary \ref{['co:iterated_pieri']}. The green chains in $2$-Bruhat order are the ones that contribute. At every level we have only recorded terms indexed by permutations in $S_7$.

Theorems & Definitions (71)

  • Theorem 1
  • Example 1.1
  • Definition 1.2
  • Theorem 1.3: Tableau criterion
  • Definition 1.4
  • Proposition 1.5
  • Definition 1.6
  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • ...and 61 more