From representations of general linear groups to plethysms of symmetric functions (and back)
17th November 2021, 2:30 pm – 3:30 pm
Fry Building, Room 2.04
Let $E$ be a $2$-dimensional vector space. The irreducible polynomial representations of the special linear group $SL_2(C)$ are the symmetric powers $Sym^r E$. Composing polynomial representations, for example to form $Sym^4 Sym^2 E$, corresponds to the plethysm product on symmetric functions. Decomposing such a representation into irreducibles, or equivalently, expressing the corresponding plethysm as a linear combinations of Schur functions, has been identified by Richard Stanley as one of the fundamental open problems in algebraic combinatorics.
In my talk I will introduce these objects and outline short proofs of some classical isomorphisms, including Hermite reciprocity $Sym^m Sym^r E \cong Sym^r Sym^m E$, and some newer isomorphisms discovered in joint work with Rowena Paget. I will then give an overview of recent results obtained with Eoghan McDowell showing that, provided suitable dualities are introduced, Hermite reciprocity holds over arbitrary fields; certain other isomorphisms (we can prove) have no generalization to prime characteristic. This opens up a new field of `modular plethysms' that we believe will repay further investigation.
My talk will assume no prior knowledge of representations of general linear groups or symmetric functions.