244 research outputs found
Enumeration of Standard Young Tableaux
A survey paper, to appear as a chapter in a forthcoming Handbook on
Enumeration.Comment: 65 pages, small correction
Crystal approach to affine Schubert calculus
We apply crystal theory to affine Schubert calculus, Gromov-Witten invariants
for the complete flag manifold, and the positroid stratification of the
positive Grassmannian. We introduce operators on decompositions of elements in
the type- affine Weyl group and produce a crystal reflecting the internal
structure of the generalized Young modules whose Frobenius image is represented
by stable Schubert polynomials. We apply the crystal framework to products of a
Schur function with a -Schur function, consequently proving that a subclass
of 3-point Gromov-Witten invariants of complete flag varieties for enumerate the highest weight elements under these operators. Included in
this class are the Schubert structure constants in the (quantum) product of a
Schubert polynomial with a Schur function for all . Another by-product gives a highest weight formulation for various fusion
coefficients of the Verlinde algebra and for the Schubert decomposition of
certain positroid classes.Comment: 42 pages; version to appear in IMR
Signed Young Modules and Simple Specht Modules
By a result of Hemmer, every simple Specht module of a finite symmetric group
over a field of odd characteristic is a signed Young module. While Specht
modules are parametrized by partitions, indecomposable signed Young modules are
parametrized by certain pairs of partitions. The main result of this article
establishes the signed Young module labels of simple Specht modules. Along the
way we prove a number of results concerning indecomposable signed Young modules
that are of independent interest. In particular, we determine the label of the
indecomposable signed Young module obtained by tensoring a given indecomposable
signed Young module with the sign representation. As consequences, we obtain
the Green vertices, Green correspondents, cohomological varieties, and
complexities of all simple Specht modules and a class of simple modules of
symmetric groups, and extend the results of Gill on periodic Young modules to
periodic indecomposable signed Young modules.Comment: To appear in Adv. Math. 307 (2017) 369--416. Proposition 4.3 (F4),
(F5) corrected, Lemma 4.9 adjusted accordingl
- …