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A new proof of an Eğecioğlu--Remmel inverse Kostka matrix problem via a Garsia--Milne involution involving Sym and NSym

Edward E. Allen, Kyle Celano, Sarah K. Mason

TL;DR

The paper develops a full combinatorial proof of the reverse Kostka identity $K^{-1}K=I$ by first establishing NSym analogues $\tilde{K}\tilde{K}^{-1}=I$ and $\tilde{K}^{-1}\tilde{K}=I$ via sign-reversing involutions on pairs of immaculate tableaux and tunnel hook coverings. A NSym–Sym reduction then recovers the classical Kostka identities in $\mathrm{Sym}$, while a Garsia–Milne involution framework provides a unified, bijective approach to the Jacobi–Trudi–type structure for immaculate functions. The authors also connect their constructions to special rim hook tableaux, establishing a bijection with tunnel hook coverings and highlighting a distinct bijection from Loehr–Mendes to a permutation model. Overall, the work presents a general, involution-based paradigm that translates signed combinatorial sums into concrete identities and suggests broader applicability to related symmetric-function problems and Jacobi–Trudi formulas.

Abstract

Eğecioğlu 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 Eğecioğlu--Remmel map. In this article, we solve the problem in full generality independent of the Eğecioğlu--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 $S_n.$

A new proof of an Eğecioğlu--Remmel inverse Kostka matrix problem via a Garsia--Milne involution involving Sym and NSym

TL;DR

The paper develops a full combinatorial proof of the reverse Kostka identity by first establishing NSym analogues and via sign-reversing involutions on pairs of immaculate tableaux and tunnel hook coverings. A NSym–Sym reduction then recovers the classical Kostka identities in , while a Garsia–Milne involution framework provides a unified, bijective approach to the Jacobi–Trudi–type structure for immaculate functions. The authors also connect their constructions to special rim hook tableaux, establishing a bijection with tunnel hook coverings and highlighting a distinct bijection from Loehr–Mendes to a permutation model. Overall, the work presents a general, involution-based paradigm that translates signed combinatorial sums into concrete identities and suggests broader applicability to related symmetric-function problems and Jacobi–Trudi formulas.

Abstract

Eğecioğlu and Remmel provide a combinatorial proof (using special rim hook tableaux) that the product of the Kostka matrix and its inverse equals the identity matrix . They then pose the problem of proving the reverse identity 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 Eğecioğlu--Remmel map. In this article, we solve the problem in full generality independent of the Eğecioğlu--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

Paper Structure

This paper contains 11 sections, 34 theorems, 73 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Background on immaculate functions and NSym Kostka matrices
  3. A combinatorial proof of Theorem \ref{['Th:One']}
  4. A combinatorial proof of Theorem \ref{['Th:Two']}
  5. A combinatorial proof of Theorem \ref{['thm:KinvK-involution']}
  6. Connections to special rim hook tableaux
  7. The permutation associated to a special rim hook tableau
  8. The relationship between the permutation and content of a special rim hook tableau
  9. Connections and comparisons
  10. Conclusion and discussion
  11. Proofs of Propositions \ref{['prop:permutation-THC-cycle-decomp']} and \ref{['prop:THC-perm-comp-row-by-row']}

Key Result

Theorem 1.1

There exists a sign-reversing involution between certain pairs of semistandard Young tableaux and special rim hook tableaux which shows combinatorially that ${\sum_{\nu\vdash n} K_{\lambda,\nu} K^{-1}_{\nu,\mu} = \delta_{\lambda,\mu}}$, where $\delta_{\lambda,\mu}$ is the Kronecker delta.

Figures (12)

  • Figure 1: The above figure is a Ferrers diagram of shape $\alpha=(2,3,4,2)$ and the corresponding collection of cells $C_\alpha.$
  • Figure 2: The permutation (in 2-line notation) associated to a tunnel hook covering
  • Figure 3: Using \ref{['alg:KKinverse-row-labeling']}, $q_1=1,q_2=2,$ and $q_3=6$. Thus $m=3$ since $\mathop{\mathrm{perm}}\nolimits(T)_1=1,\mathop{\mathrm{perm}}\nolimits(T)_2=2,$ and $\mathop{\mathrm{perm}}\nolimits(T)_6=5$. Note that row 3 is sorted into weakly increasing order when $q_3=6$ is changed into $p=3$.
  • Figure 4: Using \ref{['alg:KKinverseSym-row-labeling']}, $q_1=1,q_2=2,$ and $q_3=5$. Thus $m=3$ since $q_3 \not= 3$. Note we first change the $5$ in row 3 to a 4 and then apply Bender--Knuth, resulting in each 4 changing to 5, except the first 2 which are paired.
  • Figure 5: An example of \ref{['alg: involution']}
  • ...and 7 more figures

Theorems & Definitions (67)

  • Theorem 1.1: EgeRem90
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1: BBSSZ14
  • Theorem 2.3: AllenMason25EJC
  • Theorem 2.4: AllenMason25EJC
  • Lemma 2.5: Dominance Lemma for IT BBSSZ14
  • Lemma 2.6: Dominance Lemma for THC
  • proof
  • ...and 57 more