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Combinatorial equivalence of separable elements in types $A$ and $B$

Yong Liao, Yuping Yang, Houyi Yu

TL;DR

This work proves a tight combinatorial correspondence between type $A$ and type $B$ separable elements by constructing a bijection $\varphi_n$ from $K(S_{n+1})$ to $K(B_n)$ that preserves descent statistics and induces a left weak order isomorphism. Consequently, separable signed permutations are counted by the $n$th Schröder number, and their descent polynomials are $γ$-positive; the authors also provide explicit product formulas for the rank generating functions of principal ideals in the left weak order via separating trees. The results unify the combinatorics of separable elements across types and yield concrete enumerative and structural consequences with potential links to representation theory and root-system pattern avoidance. They extend the classical separable decomposition from type $A$ to type $B$ and establish a robust framework for further exploration of separable elements in other Weyl groups.

Abstract

We study the combinatorial equivalence of separable elements in types $A$ and $B$. A bijection is constructed from the set of separable permutations in the symmetric group $S_{n+1}$ to the set of separable signed permutations in the hyperoctahedral group $B_n$. This bijection preserves descent statistics and induces a poset isomorphism under the left weak order. As a consequence, separable signed permutations are enumerated by the large Schröder numbers, and their descent polynomials are shown to be $γ$-positive. Building on a recursive characterization of separable signed permutations via direct sum and skew sum operations, we derive explicit product formulas for the rank generating functions of the principal upper and lower ideals of separable signed permutations under the left weak order.

Combinatorial equivalence of separable elements in types $A$ and $B$

TL;DR

This work proves a tight combinatorial correspondence between type and type separable elements by constructing a bijection from to that preserves descent statistics and induces a left weak order isomorphism. Consequently, separable signed permutations are counted by the th Schröder number, and their descent polynomials are -positive; the authors also provide explicit product formulas for the rank generating functions of principal ideals in the left weak order via separating trees. The results unify the combinatorics of separable elements across types and yield concrete enumerative and structural consequences with potential links to representation theory and root-system pattern avoidance. They extend the classical separable decomposition from type to type and establish a robust framework for further exploration of separable elements in other Weyl groups.

Abstract

We study the combinatorial equivalence of separable elements in types and . A bijection is constructed from the set of separable permutations in the symmetric group to the set of separable signed permutations in the hyperoctahedral group . This bijection preserves descent statistics and induces a poset isomorphism under the left weak order. As a consequence, separable signed permutations are enumerated by the large Schröder numbers, and their descent polynomials are shown to be -positive. Building on a recursive characterization of separable signed permutations via direct sum and skew sum operations, we derive explicit product formulas for the rank generating functions of the principal upper and lower ideals of separable signed permutations under the left weak order.
Paper Structure (8 sections, 29 theorems, 101 equations, 3 figures)

This paper contains 8 sections, 29 theorems, 101 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Separable signed permutations
  4. Combinatorial equivalence between $K( S_{n+1})$ and $K( B_n)$
  5. A bijection between $K( S_{n+1})$ and $K( B_n)$
  6. Descent polynomials for separable signed permutations
  7. Poset isomorphism under the left weak order
  8. Explicit formulas for $F(\Lambda_w^B,q)$ and $F(V_w^B,q)$

Key Result

Lemma 3.1

Let $W$ be a Weyl group. If $w\in W$ is separable, then $w_0w$, $ww_0$, and $w^{-1}$ are also separable.

Figures (3)

  • Figure 1: The left weak orders on $S_4$ (left) and $B_3$ (right), where the non-seperable elements are highlighted in light gray, and the elements corresponding under the map $\varphi_3$ are labeled with the same number.
  • Figure 2: Separating trees for $1\overline{5}\,\overline{3}\,\overline{4}\,\overline{2}\,6\,\overline{9}\,\overline{7}\,\overline{8}$ and its standard permutation $846579132$.
  • Figure 3: The separating tree $T_{\overline{1}\,\overline{2}3}$.

Theorems & Definitions (50)

  • Lemma 3.1: GG20am, Corollary 2
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • proof
  • Theorem 3.6
  • proof
  • Corollary 3.7
  • ...and 40 more