Matrices connected with Brauer's centralizer algebras.

Abstract

Brauer's Centralizer Algebras were introduced by Richard Brauer (Brr) in 1937 for the purpose of studying the centralizer algebras of orthogonal and symplectic groups on the tensor powers of their defining representations. An interesting problem that has been open for many years now is to determine the algebra structure of the Brauer centralizer algebras $A\sbsp{f}{(x)}$, where f is a positive integer and the parameter x is a real number. There is an important tower of ideals $A\sbsp{f}{(x)}(0) \supseteq A\sbsp{f}{(x)}(1) \supseteq A\sbsp{f}{(x)}(2)\...$ in $A\sbsp{f}{(x)}$. One way to study Brauer's centralizer algebras is to examine the algebras $A\sbsp{f}{(x)}\lbrack k\rbrack\ = A\sbsp{f}{(x)}(k)/A\sbsp{f}{(x)}(k + 1).$ Some results about the semisimplicity of these algebras were found by Brauer, Brown and Weyl, and have been known for quite a long time (see (Brr), (Brn), and (Wl)). More recently, Hanlon and Wales (HW1) showed that the dimensions of the radical and the simple components of the semisimple part of $A\sbsp{f}{(x)}\lbrack k\rbrack $ are completely determined by the ranks of certain matrices $Y\sp{\lambda/\mu}(x)$. Finding these ranks has proved very difficult in general. The ranks have been found in several special cases, and there are many conjectures about these matrices which are supported by computational evidence. One conjecture of Hanlon and Wales was that $A\sbsp{f}{(x)}$ is semisimple unless x is a rational integer. Wenzl (Wz) used a different approach (involving "the tower construction" due to Vaughn Jones (Jo)) to prove this important result. We introduce an algebra $A\sbsp{f}{\Lambda}\lbrack k\rbrack $ with parameters $y\sb1,\...,y\sb{n}.$ If we set $y\sb{i}$ = 1 for all i, then $A\sbsp{f}{\Lambda}\lbrack k\rbrack $ is isomorphic to $A\sbsp{f}{(n)}\lbrack k\rbrack $. The algebra $A\sbsp{f}{\Lambda}\lbrack k\rbrack $ has a grading which respects the multiplication in $A\sbsp{f}{\Lambda}\lbrack k\rbrack $. This allows us to define the associated graded algebra $B\sbsp{f}{\Lambda}\lbrack k\rbrack $. One can use Hanlon and Wales' procedure to construct matrices ${\cal M}\sp{\lambda/\mu}(y\sb1,\... ,y\sb{n})$ that play the same role in $B\sbsp{f}{\Lambda}\lbrack k\rbrack $ that the matrices $Y\sp{\lambda/\mu}(x)$ play in $A\sbsp{f}{(x)}\lbrack k\rbrack $. We find an explicit formula for the determinant of ${\cal M}\sp{\lambda/\mu}(y\sb1,\... ,y\sb{n}).$ By setting $y\sb{i}$ = 1 for all i, this theorem specializes to prove a conjecture of Jockusch. The determinant of ${\cal M}\sp{\lambda/\mu}(y\sb1,\... ,y\sb{n})$ is a discriminant of the algebra $B\sbsp{f}{\Lambda}\lbrack k\rbrack $ in the sense that $B\sbsp{f}{\lambda}\lbrack k\rbrack $ is semisimple unless the n-tuple $y\sb1,\... ,y\sb{n}$ is a root of this determinant. The formula for det ${\cal M}\sp{\lambda/\mu}(y\sb1,\... ,y\sb{n})$ has an elegant statement in terms of a Jeu de Taquin algorithm defined for standard matchings. We prove a number of basic results about this new algorithm.