New approaches to the restriction problem

Abstract: Given an irreducible polynomial representation $W_n$ of the general linear group $GL_n$, we can restrict it to the representations of the symmetric group $S_n$ that seats inside $GL_n$ as a subgroup. The restriction problem is to find a combinatorial interpretation of the restriction coefficient: the multiplicity of an irreducible $S_n$ modules in such restriction of $W_n$. This is an open problem (see OPAC 2021!) in algebraic combinatorics.

In Polynomial Induction and the Restriction Problem, we construct the polynomial induction functor, which is the right adjoint to the restriction functor from the category of polynomial representations of $GL_n$ to the category of representations of $S_n$. This construction leads to a representation-theoretic proof of Littlewood's Plethystic formula for the restriction coefficient.

Character polynomials have been used to study characters of families of representations of symmetric groups (see Garsia and Goupil ), also used in the context of FI-modules by Church, Ellenberg, and Farb (see FI-modules and stability for representations of symmetric groups).

In Character Polynomials and the Restriction Problem, we compute character polynomial for the family of restrictions of $W_n$ as $n$ varies. We give an interpretation of the restriction coefficient as a moment of a certain character polynomial. To characterize partitions for which the corresponding Weyl module has non zero $S_n$-invariant vectors is quite hard. We solve this problem for partition with two rows, two columns, and for hook-partitions.

This is joint work with Sridhar Narayanan, Amritanshu Prasad, and Shraddha Srivastava.

commutative algebracombinatoricscategory theoryrepresentation theory

Audience: researchers in the topic

UC Davis algebra & discrete math seminar