2,553 research outputs found
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
Order ideals in weak subposets of Young's lattice and associated unimodality conjectures
The k-Young lattice Y^k is a weak subposet of the Young lattice containing
partitions whose first part is bounded by an integer k>0. The Y^k poset was
introduced in connection with generalized Schur functions and later shown to be
isomorphic to the weak order on the quotient of the affine symmetric group by a
maximal parabolic subgroup. We prove a number of properties for including
that the covering relation is preserved when elements are translated by
rectangular partitions with hook-length . We highlight the order ideal
generated by an rectangular shape. This order ideal, L^k(m,n),
reduces to L(m,n) for large k, and we prove it is isomorphic to the induced
subposet of L(m,n) whose vertex set is restricted to elements with no more than
k-m+1 parts smaller than m. We provide explicit formulas for the number of
elements and the rank-generating function of L^k(m,n). We conclude with
unimodality conjectures involving q-binomial coefficients and discuss how
implications connect to recent work on sieved q-binomial coefficients.Comment: 18 pages, 5 figure
Determinantal Construction of Orthogonal Polynomials Associated with Root Systems
We consider semisimple triangular operators acting in the symmetric component
of the group algebra over the weight lattice of a root system. We present a
determinantal formula for the eigenbasis of such triangular operators. This
determinantal formula gives rise to an explicit construction of the Macdonald
polynomials and of the Heckman-Opdam generalized Jacobi polynomials.Comment: 28 page
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