Irreducible calibrated representations of periplectic Brauer algebras and hook representations of the symmetric group

Abstract

We construct an infinite tower of irreducible calibrated representations of periplectic Brauer algebras on which the cup-cap generators act by nonzero matrices. As representations of the symmetric group, these are exterior powers of the standard representation (i.e. hook representations). Our approach uses the recently-defined degenerate affine periplectic Brauer algebra, which plays a role similar to that of the degenerate affine Hecke algebra in representation theory of the symmetric group. We write formulas for the representing matrices in the basis of Jucys-Murphy eigenvectors and we completely describe the spectrum of these representations. The tower formed by these representations provides a new, non-semisimple categorification of Pascal's triangle. Along the way, we also prove some basic results about calibrated representations of the degenerate affine periplectic Brauer algebra.

The Nichols algebra $\mathfrak{B}(V_{abe})$ and a class of combinatorial numbers

“…The dimensions of the irreducible submodules of N n forms exactly the Pascal's triangle according to the Proposition 2.10 and the Theorem 2.11. This is not the first time that Pascal's triangle has been categorified in a module category, please refer to [6], [7], and [10].…”