Algebra Seminar: Plethystic lifts of $q$-binomial identities

Abstract: The $q$-binomial coefficients refine the usual binomial coefficients and admit several combinatorial interpretations. For instance, they arise as the generating function for the weights of subsets, as the generating function for partitions fitting inside a rectangle, and as the number of subspaces of a vector space over $\mathbb{F}_q$. At the same time, they play a central role in representation theory, appearing as the characters of certain plethystic representations of $SL_2(\mathbb{C})$. From this perspective, every $q$-binomial identity corresponds to an isomorphism of complex representations.

A natural---but largely unexplored---question is which of these identities lift to isomorphisms of representations over finite fields, where the character theory alone no longer determines whether such algebraic lifts exist. In this talk, I will present new characteristic-free lifts of several well-known $q$-binomial identities, highlighting the novel role of symmetric polynomials as bases of representations, in contrast to their classical appearance as the characters.

Joint work with Álvaro Gutiérrez, Álvaro L. Martínez, and Mark Wildon.

This seminar is held on Zoom: https://uu-se.zoom.us/j/64555726999

This is a seminar in our algebra seminar series.