A local framework for proving combinatorial matrix inversion theorems

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 E\u{g}ecio\u{g}lu 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.