[Paper Review] Combinatorial $B_n$-analogues of Schubert polynomials

Sergey Fomin, Anatol N. Kirillov|ArXiv.org|Jun 11, 1993

Advanced Combinatorial Mathematics2 references21 citations

TL;DR

This paper introduces combinatorial $B_n$-analogues of Schubert polynomials, denoted $\goth B^{(n)}_w$, using an exponential solution of the $B_n$-Yang-Baxter equation within the nilCoxeter algebra of the hyperoctahedral group. These polynomials exhibit nonnegative coefficients, satisfy recurrence relations with divided differences (except for the special generator $s_0$), and have a direct combinatorial interpretation via $B_n$-braids and reduced decompositions. The key contribution is the construction of symmetric functions $H_w$ as a limit of $\goth B^{(n)}_w$, which generalize Schur $P$-functions and are shown to be nonnegative integer combinations of them.

ABSTRACT

Combinatorial $B_n$-analogues of Schubert polynomials and corresponding symmetric functions are constructed from an exponential solution of the $B_n$-Yang-Baxter equation that involves the nilCoxeter algebra of the hyperoctahedral group.

Motivation & Objective

• To extend the algebraic and combinatorial framework of Schubert polynomials from the symmetric group ($A_n$) to the hyperoctahedral group ($B_n$).
• To construct $B_n$-analogues of Schubert polynomials using an exponential solution of the $B_n$-Yang-Baxter equation in the nilCoxeter algebra.
• To establish a combinatorial interpretation of these polynomials in terms of $B_n$-braids and reduced decompositions.
• To define symmetric functions $H_w$ as a limit of $\goth B^{(n)}_w$, generalizing Schur $P$-functions.
• To prove that $H_w$ is a nonnegative integer linear combination of Schur $P$-functions, and that skew Schur $P$-functions arise as special cases of $H_w$.

Proposed method

• The construction uses an exponential solution of the $B_n$-Yang-Baxter equation involving formal variables and elements $h_i(x)$ in an associative algebra.
• Configurations of lines in a semi-strip with a bottom mirror are used to model $B_n$-braids, where intersection and reflection points are assigned level numbers and associated variables.
• The associated expression $\Phi(\mathcal{C})$ is built from $h_i(z)$ factors, with variables depending on segment slopes and reflection points.
• The Yang-Baxter equations impose consistency conditions on the algebraic relations of $h_i(x)$, ensuring invariance under braid moves.
• Polynomials $\goth B^{(n)}_w$ are defined via the nilCoxeter algebra, with $n^2$ factors corresponding to entries in a standard reduced decomposition of the longest element $w_0$.
• Symmetric functions $H_w$ are defined as the limit of $\goth B^{(n)}_w$ as $n \to \infty$, with $H_w$ being the stable limit of $\goth B^{(n)}_w$ under zero-padding of variables.

Experimental results

Research questions

• RQ1 How can Schubert polynomials be generalized from the symmetric group $A_n$ to the hyperoctahedral group $B_n$ using algebraic structures?
• RQ2 What combinatorial objects (e.g., braids, decompositions) correspond to the $B_n$-analogues of Schubert polynomials?
• RQ3 How do the $B_n$-Schubert polynomials relate to symmetric functions, particularly Schur $P$-functions?
• RQ4 Can the symmetric functions $H_w$ be obtained as a limit of $\goth B^{(n)}_w$, and what is the stability of their coefficients?
• RQ5 Which elements $w \in W^{(n)}$ yield $H_w = P_\sigma$, the skew Schur $P$-function for a shifted shape $\sigma$?

Key findings

• The $B_n$-Schubert polynomials $\goth B^{(n)}_w$ have nonnegative integer coefficients, as shown by their combinatorial interpretation in terms of $B_n$-braids.
• These polynomials satisfy recurrence relations with divided differences, with a special case for the generator $s_0$.
• The defining expression in the nilCoxeter algebra contains $n^2$ factors, in natural bijection with the entries of a standard reduced decomposition of the longest element $w_0$.
• The symmetric functions $H_w$ are obtained as the limit $\lim_{N\to\infty} \goth B^{(n+N)}(0^N, x_1, \dots, x_n)$, and this limit stabilizes for $N \geq n-1$.
• The functions $H_w$ are nonnegative integer linear combinations of Schur $P$-functions, and skew Schur $P$-functions arise as special cases when $w = w_\sigma$ for a skew shifted shape $\sigma$.
• The set of $w$ for which $H_w = P_\sigma$ corresponds to those avoiding the patterns $321$, $\overline{3}21$, $32\overline{1}$, $\overline{3}2\overline{1}$, $1\overline{2}$, and $\overline{1}\overline{2}$.