2,406 research outputs found
Tableaux on k+1-cores, reduced words for affine permutations, and k-Schur expansions
The -Young lattice is a partial order on partitions with no part
larger than . This weak subposet of the Young lattice originated from the
study of the -Schur functions(atoms) , symmetric functions
that form a natural basis of the space spanned by homogeneous functions indexed
by -bounded partitions. The chains in the -Young lattice are induced by a
Pieri-type rule experimentally satisfied by the -Schur functions. Here,
using a natural bijection between -bounded partitions and -cores, we
establish an algorithm for identifying chains in the -Young lattice with
certain tableaux on cores. This algorithm reveals that the -Young
lattice is isomorphic to the weak order on the quotient of the affine symmetric
group by a maximal parabolic subgroup. From this, the
conjectured -Pieri rule implies that the -Kostka matrix connecting the
homogeneous basis \{h_\la\}_{\la\in\CY^k} to \{s_\la^{(k)}\}_{\la\in\CY^k}
may now be obtained by counting appropriate classes of tableaux on -cores.
This suggests that the conjecturally positive -Schur expansion coefficients
for Macdonald polynomials (reducing to -Kostka polynomials for large )
could be described by a -statistic on these tableaux, or equivalently on
reduced words for affine permutations.Comment: 30 pages, 1 figur
Symmetric Grothendieck polynomials, skew Cauchy identities, and dual filtered Young graphs
Symmetric Grothendieck polynomials are analogues of Schur polynomials in the
K-theory of Grassmannians. We build dual families of symmetric Grothendieck
polynomials using Schur operators. With this approach we prove skew Cauchy
identity and then derive various applications: skew Pieri rules, dual
filtrations of Young's lattice, generating series and enumerative identities.
We also give a new explanation of the finite expansion property for products of
Grothendieck polynomials
Dual Filtered Graphs
We define a K-theoretic analogue of Fomin's dual graded graphs, which we call
dual filtered graphs. The key formula in the definition is DU-UD= D + I. Our
major examples are K-theoretic analogues of Young's lattice, of shifted Young's
lattice, and of the Young-Fibonacci lattice. We suggest notions of tableaux,
insertion algorithms, and growth rules whenever such objects are not already
present in the literature. We also provide a large number of other examples.
Most of our examples arise via two constructions, which we call the Pieri
construction and the Mobius construction. The Pieri construction is closely
related to the construction of dual graded graphs from a graded Hopf algebra,
as described by Bergeron-Lam-Li, Nzeutchap, and Lam-Shimizono. The Mobius
construction is more mysterious but also potentially more important, as it
corresponds to natural insertion algorithms.Comment: 54 pages, small edits made in new versio
Lattice Diagram Polynomials and Extended Pieri Rules
The lattice cell in the row and column of the
positive quadrant of the plane is denoted . If is a partition of
, we denote by the diagram obtained by removing the cell
from the (French) Ferrers diagram of . We set , where are the
cells of , and let be the linear span of the partial
derivatives of . The bihomogeneity of and
its alternating nature under the diagonal action of gives the structure of a bigraded -module. We conjecture that is always a direct sum of left regular representations of
, where is the number of cells that are weakly north and east of
in . We also make a number of conjectures describing the precise
nature of the bivariate Frobenius characteristic of in terms
of the theory of Macdonald polynomials. On the validity of these conjectures,
we derive a number of surprising identities. In particular, we obtain a
representation theoretical interpretation of the coefficients appearing in some
Macdonald Pieri Rules.Comment: 77 pages, Te
Quantum Integrals for a Semi-Infinite -Boson System with Boundary Interactions
We provide explicit formulas for the quantum integrals of a semi-infinite
-boson system with boundary interactions. These operators and their
commutativity are deduced from the Pieri formulas for a
Hall-Littlewood type degeneration of the Macdonald-Koornwinder polynomials
- …