Powersum Functions in the Hopf Monoid of Supercharacter Functions on UTn(q)

Undergraduate Honors Thesis

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Citeable URL: https://scholar.colorado.edu/concern/undergraduate_honors_theses/rn301209f

Abstract

The character theory of the group UTn(q) of unipotent upper triangular matrices over a ﬁnite ﬁeld of order q is known to be wild. However, in a generalization of character theory called supercharacter theory, one ﬁnds that there is a connection between the representation theory of UTn(q), the combinatorics of set partitions, and the algebra of symmetric function in non-commuting variables. The relationship is reminiscent of the relationship between the symmetric group Sn, integer partitions and the algebra of symmetric functions.In this thesis I begin by giving a brief review of representation theory and Hopf monoids. I then introduce a particular supercharacter theory of UTn(q) and its connection to set partitions. A Hopf monoid is then constructed out of supercharacters of the inﬁnite family of groups, UTn(q), and the powersum basis of this Hopf monoid is reviewed. The product, coproduct, pointwise product, and antipode are then computed for the powersum basis and a q-deformation of this basis.

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