Jing's Hall-Littlewood vertex operators and generalized Kostka polynomials

Mark Shimozono

Virginia Tech

March 15,
refreshments at 3:45pm


Jing defined some linear operators on the space of symmetric functions using so-called vertex operators. He showed that certain compositions of such operators, when acting on the symmetric function 1, create the Hall-Littlewood symmetric functions. Using this he gave formulas for the Kostka-Foulkes 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 so-called 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 K-theory 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).

Speaker's Contact Info: mshimo(at-sign)calvin.math.vt.edu

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