A local framework for proving combinatorial matrix inversion theorems
Aditya Khanna, Nicholas A. Loehr
TL;DR
The paper introduces a universal local framework for proving combinatorial matrix inversions by reducing AB = I to a local identity derived from incremental object constructions such as horizontal strips, rim-hooks, or bricks. It formalizes recursions for two matrix families (A_n,B_n) indexed by compositions and shows how local weight identities imply global inverses, with a sorting-enforced pathway to square inverses. Four classical applications—rectangular Kostka matrices, rim-hook matrices, composition-refinement incidence matrices, and brick tabloids—illustrate the method and yield new bijective proofs or streamlined constructions, including a canonical bijection for rectangular Kostka inverses and a shorter bijective derivation for symmetric-group character orthogonality variants. The framework thus provides a systematic route to invert a wide class of combinatorial transition matrices and to automatically construct bijective proofs of AB = I in several settings, linking symmetric function theory with explicit combinatorial bijections.
Abstract
Combinatorial transition matrices arise frequently in the theory of symmetric functions and their generalizations. The entries of such matrices often count signed, weighted combinatorial structures such as semistandard tableaux, rim-hook tableaux, or brick tabloids. Bijective proofs that two such matrices are inverses of each other may be difficult to find. This paper presents a general framework for proving such inversion results in the case where the combinatorial objects are built up recursively by successively adding some incremental structure such as a single horizontal strip or rim-hook. In this setting, we show that a sequence of matrix inversion results $A_nB_n=I$ can be reduced to a certain ``local'' identity involving the incremental structures. Here, $A_n$ and $B_n$ are matrices that might be non-square, and the columns of $A_n$ and the rows of $B_n$ indexed by compositions of $n$. We illustrate the general theory with four classical applications involving the Kostka matrices, the character tables of the symmetric group, incidence matrices for composition posets, and matrices counting brick tabloids. We obtain a new, canonical bijective proof of an inversion result for rectangular Kostka matrices, which complements the proof for the square case due to Eğecioğlu and Remmel. We also give a new bijective proof of the orthogonality result for the irreducible $S_n$-characters that is shorter than the original version due to White.