Benjamin Morris (University of Leeds) – Semi simplicity criterion for the Kadar-Yu algebras

In a 2019 paper, Kadar, Martin, and Yu, introduced the notion of left height for a Brauer diagram, and used it to define a family of subalgebras of the Brauer algebra which interpolate between the Temperley-Lieb (sub)algebra, and the full Brauer Algebra. Early calculations indicated that for intermediate height values, these algebras exhibited novel semi-simplicity criteria distinct from the classical criteria in the Temperley-Lieb case (roots of unity), and Brauer case (integers). In this talk, we present ongoing work aimed characterising these criteria; rules for computing Gram-matrix determinants for all standard modules are given in some low height cases, leading to a general conjecture. This conjecture involves introducing a Chebyshev series of polynomials for each integer partition, the roots of which seem to “resemble” roots of unity for large n. Time permitting, we will discuss some of the representation theoretic consequences of our calculations for the Kadar-Yu algebras in low rank.