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Type C $K$-Stanley symmetric functions and Kraśkiewicz-Hecke insertion

Joshua Arroyo, Zachary Hamaker, Graham Hawkes, Jianping Pan

TL;DR

This work develops a Type C $K$-theoretic framework for Stanley symmetric functions, introducing $G^C_w$ indexed by signed permutations and their conjectural expansion into shifted Grothendieck analogues $GQ^{(\beta)}_\lambda$. Central to the approach is Kraśkiewicz--Hecke (KH) insertion, a new insertion algorithm that merges Kraśkiewicz and Hecke insertions and yields a weight-preserving bijection between words and pairs consisting of a strict decomposition tableau and a shifted set-valued recording tableau. The authors establish two crucial special cases: (i) for vexillary signed permutations, $G^C_w$ equals a single shifted $GQ^{(\beta)}_\lambda$, and (ii) for certain skew shapes, $G^C_w$ equals a single skew $GQ^{(\beta)}_{\lambda/\mu}$, thereby giving explicit combinatorial descriptions of corresponding $GQ$ expansions. They propose Conjecture 6.1 predicting a positive $GQ^{(\beta)}$-expansion with coefficients counting strict decomposition tableaux whose reading words are $0$-Hecke expressions, and they derive consequences for products involving trapezoidal shapes. The work bridges combinatorics of shifted tableaux with $K$-theory of the Lagrangian Grassmannian and outlines future directions for establishing full positivity rules and product formulas in Type C.

Abstract

We study Type C $K$-Stanley symmetric functions, which are $K$-theoretic extensions of the Type C Stanley symmetric functions. They are indexed by signed permutations and can be used to enumerate reduced words via their expansion into Schur $Q$-functions, which are indexed by strict partitions. A combinatorial description of the Schur $Q$- coefficients is given by Kraśkiewicz insertion. Similarly, their $K$-Stanley analogues are conjectured to expand positively into $GQ$'s, which are $K$-theory representatives for the Lagrangian Grassmannian introduced by Ikeda and Naruse also indexed by strict partitions. We introduce a $K$-theoretic analogue of Kraśkiewicz insertion, which can be used to enumerate 0-Hecke expressions for signed permutations and gives a conjectural combinatorial rule for computing this $GQ$ expansion. We show the Type C $K$-Stanleys for certain fully commutative signed permutations are skew $GQ$'s. Combined with a Pfaffian formula of Anderson's, this allows us to prove Lewis and Marberg's conjecture that $GQ$'s of (skew) rectangle shape are $GQ$'s of trapezoid shape. Combined with our previous conjecture, this also gives an explicit combinatorial description of the skew $GQ$ expansion into $GQ$'s. As a consequence, we obtain a conjecture for the product of two $GQ$ functions where one has trapezoid shape.

Type C $K$-Stanley symmetric functions and Kraśkiewicz-Hecke insertion

TL;DR

This work develops a Type C -theoretic framework for Stanley symmetric functions, introducing indexed by signed permutations and their conjectural expansion into shifted Grothendieck analogues . Central to the approach is Kraśkiewicz--Hecke (KH) insertion, a new insertion algorithm that merges Kraśkiewicz and Hecke insertions and yields a weight-preserving bijection between words and pairs consisting of a strict decomposition tableau and a shifted set-valued recording tableau. The authors establish two crucial special cases: (i) for vexillary signed permutations, equals a single shifted , and (ii) for certain skew shapes, equals a single skew , thereby giving explicit combinatorial descriptions of corresponding expansions. They propose Conjecture 6.1 predicting a positive -expansion with coefficients counting strict decomposition tableaux whose reading words are -Hecke expressions, and they derive consequences for products involving trapezoidal shapes. The work bridges combinatorics of shifted tableaux with -theory of the Lagrangian Grassmannian and outlines future directions for establishing full positivity rules and product formulas in Type C.

Abstract

We study Type C -Stanley symmetric functions, which are -theoretic extensions of the Type C Stanley symmetric functions. They are indexed by signed permutations and can be used to enumerate reduced words via their expansion into Schur -functions, which are indexed by strict partitions. A combinatorial description of the Schur - coefficients is given by Kraśkiewicz insertion. Similarly, their -Stanley analogues are conjectured to expand positively into 's, which are -theory representatives for the Lagrangian Grassmannian introduced by Ikeda and Naruse also indexed by strict partitions. We introduce a -theoretic analogue of Kraśkiewicz insertion, which can be used to enumerate 0-Hecke expressions for signed permutations and gives a conjectural combinatorial rule for computing this expansion. We show the Type C -Stanleys for certain fully commutative signed permutations are skew 's. Combined with a Pfaffian formula of Anderson's, this allows us to prove Lewis and Marberg's conjecture that 's of (skew) rectangle shape are 's of trapezoid shape. Combined with our previous conjecture, this also gives an explicit combinatorial description of the skew expansion into 's. As a consequence, we obtain a conjecture for the product of two functions where one has trapezoid shape.

Paper Structure

This paper contains 14 sections, 29 theorems, 67 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Stanley symmetric functions and Kraśkiewicz--Hecke insertion
  3. Background
  4. Signed Permutations
  5. Shifted Tableaux
  6. Type $C$ $K$--Stanley symmetric functions
  7. Definition and basic properties
  8. Top fully commutative elements
  9. Strict Decomposition Tableaux
  10. Kraśkiewicz--Hecke Insertion
  11. Insertion
  12. Inverse insertion
  13. Conjectures
  14. Concluding Remarks

Key Result

Theorem 1.1

For all $n \in \mathbb{N}$, the map Kraśkiewicz--Hecke insertion is a bijection: Moreover, for $KH(a_1,\dots,a_p) = (P,Q)$, the words $(a_1,\dots,a_p)$ and $\rho(P)$ are $0$--Hecke expressions for the same signed permutation.

Figures (1)

  • Figure 1: To the left is a shifted Young diagram of shape $\lambda=(4,3,1)$. Next is a standard shifted Young tableau $T_1$ of the same shape with peak set $\mathsf{Peak}(T_1) = \{2,5\}$, followed by a standard shifted set valued Young tableau $T_2$ of the same shape with $\mathsf{Peak}(T_2) = \{4,6,8,10\}$. Last is a semistandard shifted set valued Young tableau $T_3$ so that $\mathsf{st}(T_3) = T_2$.

Theorems & Definitions (71)

  • Theorem 1.1
  • Corollary 1.2
  • Conjecture 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 3.1
  • proof
  • Proposition 3.2: kirillov2017construction
  • proof
  • Corollary 3.3
  • ...and 61 more