Combinatorial Invariance of Kazhdan-Lusztig-Vogan Polyomials for Fixed Point Free Involutions

TL;DR

This paper proves that Kazhdan-Lusztig-Vogan polynomials associated with fixed point free involutions are combinatorial invariants under Bruhat order isomorphisms, revealing a deep symmetry in their structure.

Contribution

It establishes that these polynomials remain unchanged under poset isomorphisms of Bruhat intervals, highlighting their combinatorial invariance.

Findings

Kazhdan-Lusztig-Vogan polynomials are invariant under Bruhat order isomorphisms.

The invariance applies specifically to fixed point free involutions in symmetric groups.

The result links geometric orbit structure with combinatorial properties of polynomials.

Abstract

When $Sp(2n,\mathbb{C})$ acts on the flag variety of $SL(2n,\mathbb{C})$, the orbits are in bijection with fixed point free involutions in the symmetric group $S_{2n}$. In this case, the associated Kazhdan-Lusztig-Vogan polynomials $P_{v,u}$ can be indexed by pairs of fixed point free involutions $v\geq u$, where $\geq$ denotes the Bruhat order on $S_{2n}$. We prove that these polynomials are combinatorial invariants in the sense that if $f: [u, w_0 ] \rightarrow [u , w_0]$ is a poset isomorphism of upper intervals in the Bruhat order on fixed point free involutions, then $P_{v,u} = P_{f(v),u}$ for all $v \geq u$.