A skein model for unipotent Hecke algebrasNat TiemUniversity of Wisconsin, Madison
November 19,

ABSTRACT


The classical IwahoriHecke 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 IwahoriHecke algebra by substituting the unipotent subgroup $U$ (uppertriangular matrices with ones on the diagonal) for the Borel subgroup $B$ (uppertriangular 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 RSKinsertion correspondence that associates monomial matrices to pairs of column strict multitableaux. 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. 
Combinatorics Seminar, Mathematics Department, MIT, sara(atsign)math.mit.edu 

