The partition algebra arises naturally out of the representation theory of the symmetric group and the -fold tensor of its permutation module. I'll show how to construct the partition algebra by defining a multiplication of diagrams, and then discuss some interesting subalgebras. In exploring representations of the partition algebra we'll discover several connections between the partition algebra and the symmetric group.
Partition Algebras and the Symmetric Group
Apr. 17, 2012 2pm (MATH 350)
Lie Theory
Scott Andrews (CU)
X
Determining the irreducible characters of the group of unipotent upper triangular matrices over a finite field is a provably difficult problem. Kirillov constructed class functions of this group which he conjectured to be the irreducible characters. Although he was incorrect, the Kirillov functions have important applications to the study of the character theory of this group, as well as generalizations in other types. I will construct the Kirillov functions in type A and investigate their application in other types.
Kirillov Functions
May. 01, 2012 2pm (MATH 350)
Lie Theory
Rob Maier (U. Arizona)
X
Coxeter groups occur as both automorphism groups and monodromy groups of certain Fuchsian differential equations. This includes the Picard-Fuchs equations associated to families of elliptic curves (i.e., associated to elliptic surfaces). Such equations are second-order differential equations that are extensions of the Gauss hypergeometric equation (e.g., they may be of the Heun type). The appearance of Coxeter groups as their automorphism groups will be explained, and it will also be briefly explained how Coxeter groups appear as the monodromy groups of certain generalized hypergeometric equations (those with algebraic solutions), as has been known since work of Frits Beukers.
Partition Algebras and the Symmetric Group