205 research outputs found
On some noncommutative symmetric functions analogous to Hall-Littlewood and Macdonald polynomials
We investigate the connections between various noncommutative analogues of
Hall-Littlewood and Macdonald polynomials, and define some new families of
noncommutative symmetric functions depending on two sequences of parameters.Comment: 20 page
q and q,t-Analogs of Non-commutative Symmetric Functions
We introduce two families of non-commutative symmetric functions that have
analogous properties to the Hall-Littlewood and Macdonald symmetric functions.Comment: Different from analogues in math.CO/0106191 - v2: 26 pages - added a
definition in terms of triangularity/scalar product relations - to be
submitted FPSAC'0
Transition matrices for symmetric and quasisymmetric Hall-Littlewood polynomials
We introduce explicit combinatorial interpretations for the coefficients in
some of the transition matrices relating to skew Hall-Littlewood polynomials
P_lambda/mu(x;t) and Hivert's quasisymmetric Hall-Littlewood polynomials
G_gamma(x;t). More specifically, we provide: 1) the G-expansions of the
Hall-Littlewood polynomials P_lambda, the monomial quasisymmetric polynomials
M_alpha, the quasisymmetric Schur polynomials S_alpha, and the peak
quasisymmetric functions K_alpha; 2) an expansion of P_lambda/mu in terms of
the F_alpha's. The F-expansion of P_lambda/mu is facilitated by introducing
starred tableaux.Comment: 28 pages; added brief discussion of the Hall-Littlewood Q', typos
corrected, added references in response to referee suggestion
Cylindric versions of specialised Macdonald functions and a deformed Verlinde algebra
We define cylindric generalisations of skew Macdonald functions when one of
their parameters is set to zero. We define these functions as weighted sums
over cylindric skew tableaux: fixing two integers n>2 and k>0 we shift an
ordinary skew diagram of two partitions, viewed as a subset of the
two-dimensional integer lattice, by the period vector (n,-k). Imposing a
periodicity condition one defines cylindric skew tableaux as a map from the
periodically continued skew diagram into the integers. The resulting cylindric
Macdonald functions appear in the coproduct of a commutative Frobenius algebra,
which is a particular quotient of the spherical Hecke algebra. We realise this
Frobenius algebra as a commutative subalgebra in the endomorphisms over a
Kirillov-Reshetikhin module of the quantum affine sl(n) algebra. Acting with
special elements of this subalgebra, which are noncommutative analogues of
Macdonald polynomials, on a highest weight vector, one obtains Lusztig's
canonical basis. In the limit q=0, one recovers the sl(n) Verlinde algebra,
i.e. the structure constants of the Frobenius algebra become the WZNW fusion
coefficients which are known to be dimensions of moduli spaces of generalized
theta-functions and multiplicities of tilting modules of quantum groups at
roots of unity. Further motivation comes from exactly solvable lattice models
in statistical mechanics: the cylindric Macdonald functions arise as partition
functions of so-called vertex models obtained from solutions to the quantum
Yang-Baxter equation. We show this by stating explicit bijections between
cylindric tableaux and lattice configurations of non-intersecting paths. Using
the algebraic Bethe ansatz the idempotents of the Frobenius algebra are
computed.Comment: 77 pages, 12 figures; v3: some minor typos corrected and title
slightly changed. Version to appear in Comm. Math. Phy
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
-Schur functions and affine Schubert calculus
This book is an exposition of the current state of research of affine
Schubert calculus and -Schur functions. This text is based on a series of
lectures given at a workshop titled "Affine Schubert Calculus" that took place
in July 2010 at the Fields Institute in Toronto, Ontario. The story of this
research is told in three parts: 1. Primer on -Schur Functions 2. Stanley
symmetric functions and Peterson algebras 3. Affine Schubert calculusComment: 213 pages; conference website:
http://www.fields.utoronto.ca/programs/scientific/10-11/schubert/, updates
and corrections since v1. This material is based upon work supported by the
National Science Foundation under Grant No. DMS-065264
A Combinatorial Derivation of the Racah-Speiser Algorithm for Gromov-Witten invariants
Using a finite-dimensional Clifford algebra a new combinatorial product
formula for the small quantum cohomology ring of the complex Grassmannian is
presented. In particular, Gromov-Witten invariants can be expressed through
certain elements in the Clifford algebra, this leads to a q-deformation of the
Racah-Speiser algorithm allowing for their computation in terms of Kostka
numbers. The second main result is a simple and explicit combinatorial formula
for projecting product expansions in the quantum cohomology ring onto the sl(n)
Verlinde algebra. This projection is non-trivial and amounts to an identity
between numbers of rational curves intersecting Schubert varieties and
dimensions of moduli spaces of generalised theta-functions.Comment: 24 pages, 3 figure
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