Jing's HallLittlewood vertex operators and generalized Kostka polynomialsMark ShimozonoVirginia Tech
March 15,

ABSTRACT


Jing defined some linear operators on the space of symmetric functions using socalled vertex operators. He showed that certain compositions of such operators, when acting on the symmetric function 1, create the HallLittlewood symmetric functions. Using this he gave formulas for the KostkaFoulkes polynomials, which are the change of basis matrix between the HL basis and the modified Schur function basis. We define operators that give rise in a similar manner, to the socalled generalized Kostka polynomials, which are defined as graded multiplicities of isotypic components of the Euler characteristic characters of some nice modules supported in the closure of a nilpotent conjugacy class. Using the vertex operators one may easily derive various new relations among the above characters and hence among the "genKostkas". Finally, one may realize as a graded R(GL(n))module, the (GL(n) x C^*)equivariant Ktheory of the nullcone, as a vector space spanned by certain of these vertex operators. This also works for other nilpotent conjugacy class closures in gl(n). 
Combinatorics Seminar, Mathematics Department, MIT, sara(atsign)math.mit.edu 

