425 research outputs found
A Bijective Proof of a Factorization Formula for Specialized Macdonald Polynomials
Let μ and ν = (ν 1, . . . , ν k ) be partitions such that μ is obtained from ν by adding m parts of sizer. Descouens and Morita proved algebraically that the modified Macdonald polynomials
H~μ(X;q,t)
satisfy the identity
H~μ=H~νH~(rm)
when the parameter t is specialize to an mth root of unity. Descouens, Morita, and Numata proved this formula bijectively when r ≤ ν k and
r∈{1,2}.
This note gives a bijective proof of the formula for all r ≤ ν k
A Markov growth process for Macdonald's distribution on reduced words
We give an algorithmic-bijective proof of Macdonald's reduced word identity
in the theory of Schubert polynomials, in the special case where the
permutation is dominant. Our bijection uses a novel application of David
Little's generalized bumping algorithm. We also describe a Markov growth
process for an associated probability distribution on reduced words. Our growth
process can be implemented efficiently on a computer and allows for fast
sampling of reduced words. We also discuss various partial generalizations and
links to Little's work on the RSK algorithm.Comment: 16 pages, 5 figure
Factorization theorems for classical group characters, with applications to alternating sign matrices and plane partitions
We show that, for a certain class of partitions and an even number of
variables of which half are reciprocals of the other half, Schur polynomials
can be factorized into products of odd and even orthogonal characters. We also
obtain related factorizations involving sums of two Schur polynomials, and
certain odd-sized sets of variables. Our results generalize the factorization
identities proved by Ciucu and Krattenthaler (Advances in combinatorial
mathematics, 39-59, 2009) for partitions of rectangular shape. We observe that
if, in some of the results, the partitions are taken to have rectangular or
double-staircase shapes and all of the variables are set to 1, then
factorization identities for numbers of certain plane partitions, alternating
sign matrices and related combinatorial objects are obtained.Comment: 22 pages; v2: minor changes, published versio
Monotone Hurwitz numbers in genus zero
Hurwitz numbers count branched covers of the Riemann sphere with specified
ramification data, or equivalently, transitive permutation factorizations in
the symmetric group with specified cycle types. Monotone Hurwitz numbers count
a restricted subset of the branched covers counted by the Hurwitz numbers, and
have arisen in recent work on the the asymptotic expansion of the
Harish-Chandra-Itzykson-Zuber integral. In this paper we begin a detailed study
of monotone Hurwitz numbers. We prove two results that are reminiscent of those
for classical Hurwitz numbers. The first is the monotone join-cut equation, a
partial differential equation with initial conditions that characterizes the
generating function for monotone Hurwitz numbers in arbitrary genus. The second
is our main result, in which we give an explicit formula for monotone Hurwitz
numbers in genus zero.Comment: 22 pages, submitted to the Canadian Journal of Mathematic
Factorization of the canonical bases for higher level Fock spaces
The level l Fock space admits canonical bases G_e and G_\infty. They
correspond to U_{v}(hat{sl}_{e}) and U_{v}(sl_{\infty})-module structures. We
establish that the transition matrices relating these two bases are
unitriangular with coefficients in N[v]. Restriction to the highest weight
modules generated by the empty l-partition then gives a natural quantization of
a theorem by Geck and Rouquier on the factorization of decomposition matrices
which are associated to Ariki-Koike algebras.Comment: The last version generalizes and proves the main conjecture of the
previous one. Final versio
A bijective proof of Macdonald's reduced word formula
International audienceWe describe a bijective proof of Macdonald's reduced word identity using pipe dreams and Little's bumping algorithm. The proof extends to a principal specialization of the identity due to Fomin and Stanley. Our bijective tools also allow us to address a problem posed by Fomin and Kirillov from 1997, using work of Wachs, Lenart and Serrano- Stump
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
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