Unipotent Hecke algebras
Nat Thiem
Stanford University
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, describe a skein model for multiplying these
basis elements (analogous to the braid relations above), and allow their
representation theory to inspire 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.