Andrew Berget

A short proof of Gamas’s theorem

Abstract If χ λ is the irreducible character of S n corresponding to the partition λ of n then we may symmetrize a tensor v 1 ⊗ ⋯ ⊗ v n by χ λ . Gamas’s theorem states that the result is not zero if and only if we can partition the set { v i } into linearly independent sets whose sizes are the parts of the transpose of λ . We give a short and self-contained proof of this fact.

Constructions for cyclic sieving phenomena

We show how to derive new instances of the cyclic sieving phenomenon from old ones via elementary representation theory. Examples are given involving objects such as words, parking functions, finite fields, and graphs.

EQUIVARIANT CHOW CLASSES OF MATRIX ORBIT CLOSURES

Let G be the product GLr(C) × (C×)n. We show that the G-equivariant Chow class of a G orbit closure in the space of r-by-n matrices is determined by a matroid. To do this, we split the natural surjective map from the G equvariant Chow ring of the space of matrices to the torus equivariant Chow ring of the Grassmannian. The splitting takes the class of a Schubert variety to the corresponding factorial Schur polynomial, and also has the property that the class of a subvariety of the Grassmannian is mapped to the class of the closure of those matrices whose row span is in the variety.

Equality of symmetrized tensors and the flag variety

The goal of this note is to give a transparent proof of a result of da Cruz and Dias da Silva on the equality of symmetrized decomposable tensors. This will be done by explaining that their result follows from the fact that the coordinate ring of a flag variety is a unique factorization domain. Let λ be a partition of a positive integer n and let χ be the irreducible character of the symmetric group Sn corresponding to λ. There is a right action of Sn on V , where V is a finite-dimensional complex vector space, by permuting tensor positions. Let Tλ be the endomorphism of V ⊗n given by

Critical groups of graphs with reflective symmetry

The critical group of a graph is a finite Abelian group whose order is the number of spanning forests of the graph. For a graph G with a certain reflective symmetry, we generalize a result of Ciucu–Yan–Zhang factorizing the spanning tree number of G by interpreting this as a result about the critical group of G. Our result takes the form of an exact sequence, and explicit connections to bicycle spaces are made.

Matrix orbit closures

Let G be the group

Products of linear forms and Tutte polynomials

\mathrm {GL}_r(\mathbf {C}) \times (\mathbf {C}^\times )^n.

Extending the parking space

GLr(C)×(C×)n. We conjecture that the finely-graded Hilbert series of a G orbit closure in the space of r-by-n matrices is wholly determined by the associated matroid. In support of this, we prove that the coefficients of this Hilbert series corresponding to certain hook-shaped Schur functions in the