A skein model for unipotent Hecke algebras

Nat Tiem

University of Wisconsin, Madison

November 19,
refreshments at 3:45pm


The classical Iwahori-Hecke algebra for $G=GL_n(F_q)$ can be thought of as a $q$-analog of the symmetric group $S_n$. It has generators indexed by simple reflections, relations that generalize the symmetric group's braid relations, and a representation theory with a strong connection to Young tableaux. Unipotent Hecke algebras are a family of Hecke algebras that generalize the Iwahori-Hecke algebra by substituting the unipotent subgroup $U$ (upper-triangular matrices with ones on the diagonal) for the Borel subgroup $B$ (upper-triangular matrices) in the Hecke algebra construction.

This talk defines unipotent Hecke algebras and analyzes some combinatorial implications of their construction. In particular, I construct an explicit basis, and describe a skein model for multiplying these basis elements (analogous to the braid relations above). Finally, the representation theory of unipotent Hecke algebras leads to a generalization of the RSK-insertion correspondence that associates monomial matrices to pairs of column strict multi-tableaux. While a large portion of this talk addresses the $G=GL_n(F_q)$ case, the unipotent Hecke algebra construction and skein model generalize to arbitrary finite groups of Lie type.

Speaker's Contact Info: thiem(at-sign)math.wisc.edu

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Combinatorics Seminar, Mathematics Department, MIT, sara(at-sign)math.mit.edu

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