Speaker
Description
Let $A = (A_n)_{n\geq 0}$ and $B = (B_n)_{n\geq 0}$ be families of "recursive'' rectangular matrices, that is, the entries of $A_n$ can be expressed as a linear combination of entries of $A_m$ for $m < n$, and similarly for $B$. Such families arise in algebraic combinatorics as change-of-basis matrices between the basis of symmetric functions and their generalizations. The entries of such matrices can be computed as sums over signed-weighted combinatorial objects which themselves can be constructed recursively. In this talk, I will describe a framework that reduces the condition $A_nB_n = I$ for all $n\geq 0$ on the matrix families to an equivalent local condition on the objects which can be handled using combinatorial methods such as sign-reversing involutions. This framework has applications in producing canonical bijective proofs for the Kostka matrix case, the orthogonality of symmetric group characters, and Mobius inversion for the lattice of sets. All combinatorial concepts will be defined in the talk.