Annihilating fields of standard modules of sl(2,C)~ and combinatorial identities

We show that a set of local admissible fields generates a vertex algebra. For an affine Lie algebra \tilde\goth g we construct the corresponding level $k$ vertex operator algebra and we show that level $k$ highest weight \tilde\goth g-modules are modules for this vertex operator algebra. We determine the set of annihilating fields of level $k$ standard modules and we study the corresponding loop \tilde\goth g module---the set of relations that defines standard modules. In the case when \tilde\goth g is of type $A_1^{(1)}$, we construct bases of standard modules parameterized by colored partitions and, as a consequence, we obtain a series of Rogers-Ramanujan type combinatorial identities.Comment: 83 pages, amste

Mask formulas for cograssmannian Kazhdan-Lusztig polynomials

We give two contructions of sets of masks on cograssmannian permutations that can be used in Deodhar's formula for Kazhdan-Lusztig basis elements of the Iwahori-Hecke algebra. The constructions are respectively based on a formula of Lascoux-Schutzenberger and its geometric interpretation by Zelevinsky. The first construction relies on a basis of the Hecke algebra constructed from principal lower order ideals in Bruhat order and a translation of this basis into sets of masks. The second construction relies on an interpretation of masks as cells of the Bott-Samelson resolution. These constructions give distinct answers to a question of Deodhar.Comment: 43 page