SEMINAR: Groups and Combinatorics Seminar 16:00 Fri 17/11/2017 Weatherburn LT: Stephen Glasby (UWA)

There will be refreshments in Maths & Stats tea room at 15:40, and we will go for drinks at the UWA Tavern or the UniClub after the seminar. All are most welcome.

Speaker: Stephen Glasby (University of Western Australia)

Title: Norman involutions and tensor products of unipotent Jordan blocks

Time and place: 16:00 Friday 17/11/2017 in Weatherburn LT

Abstract: Suppose R is an rxr unipotent matrix over some field F, i.e. its characteristic polynomial is (t-1)^r. The Jordan form of R is a sum of unipotent Jordan blocks, so we obtain some partition of r. If S is a unipotent sxs matrix over F, then so is R times S. To understand the partition of rs afforded by R times S it suffices to understand the partition afforded by J_r times J_s where J_r denotes a single rxr unipotent Jordan block. When char(F)=p, we denote this partition by lambda(r,s,p).

When p>0, the partitions lambda(r,s,p) are shrouded in mystery. Assume r<= s. We show that there is a larger set of 2^{r-1}-1 partitions (which is independent of p) and contains the mysterious partitions. These partitions correspond to involutions in the symmetric group S_r of degree r, and also to nonempty subsets of the set {1,2,...,r-1}. We also show that the group G(r,p)=<lambda(r,s,p) | s>= r>, is a wreath product, and we determine its structure.

One motivation for this research comes from representation theory: understanding the structure of the Green ring. This is joint work with Cheryl E. Praeger and Binzhou Xia.