A new proof of an E\u{g}ecio\u{g}lu--Remmel inverse Kostka matrix problem via a Garsia--Milne involution involving Sym and NSym

TL;DR

This paper provides a comprehensive combinatorial proof of the inverse Kostka matrix identity using involutions in noncommutative symmetric functions, extending previous partial results and establishing a new bijection distinct from prior work.

Contribution

It introduces a novel combinatorial proof for the inverse Kostka matrix identity using sign-reversing involutions in NSym, independent of earlier bijections.

Findings

Proved NSym versions of Kostka matrix identities.

Constructed sign-reversing involutions involving tunnel hook coverings.

Established a new injective map between rim tableaux and the symmetric group.

Abstract

E\u{g}ecio\u{g}lu and Remmel provide a combinatorial proof (using special rim hook tableaux) that the product of the Kostka matrix $K$ and its inverse $K^{-1}$ equals the identity matrix $I$. They then pose the problem of proving the reverse identity $K^{-1}K =I$ combinatorially. Sagan and Lee prove a special case of this identity using overlapping special rim hook tableaux. Loehr and Mendes provide a full proof using bijective matrix algebra that relies on the E\u{g}ecio\u{g}lu--Remmel map. In this article, we solve the problem in full generality independent of the E\u{g}ecio\u{g}lu--Remmel bijection. To do this, we start by proving NSym versions of both Kostka matrix identities using sign-reversing involutions involving the tunnel hook coverings recently introduced by the first and third authors. Then we modify our sign-reversing involutions to reduce to Sym. Finally, we show that our bijection is different than the Loehr and Mendes result by constructing an injective map between special rim tableaux and the symmetric group $S_n.$