A Combinatorial Formula for Affine Hall-Littlewood Functions via a Weighted Brion Theorem

B. L. Feigin; I. Makhlin

?

A Combinatorial Formula for Affine Hall-Littlewood Functions via a Weighted Brion Theorem

2015.

We present a new combinatorial formula for Hall-Littlewood functions associated with the affine root system of type A~n−1, i.e. corresponding to the affine Lie algebra slˆn. Our formula has the form of a sum over the elements of a basis constructed by Feigin, Jimbo, Loktev, Miwa and Mukhin in the corresponding irreducible representation.Our formula can be viewed as a weighted sum of exponentials of integer points in a certain infinite-dimensional convex polyhedron. We derive a weighted version of Brion's theorem and then apply it to our polyhedron to prove the formula.

Language: English

Text on another site

Keywords: Gelfand-Tsetlin polytope Brion's theorem Hall-Littlewood functions convex polyhedra Affine Lie algebra

Stochastic local volatility models and the Wei-Norman factorization method

Guerrero J., Orlando G., Discrete and Continuous Dynamical Systems - Series S 2022 Vol. 15 No. 12 P. 3699–3722

In this paper, we show that a time-dependent local stochastic volatility (SLV) model can be reduced to a system of autonomous PDEs that can be solved using the heat kernel, by means of the Wei-Norman factorization method and Lie algebraic techniques. Then, we compare the results of traditional Monte Carlo simulations with the explicit solutions ...

Added: February 23, 2024

Beilinson-Drinfeld Schubert varieties and global Demazure modules

Dumanski I., Feigin E., Finkelberg M. V., / Series math "arxiv.org". 2020. No. 2003.12930.

We compute the spaces of sections of powers of the determinant line bundle on the spherical Schubert subvarieties of the Beilinson- Drinfeld affine Grassmannians. The answer is given in terms of global Demazure modules of the current Lie algebra. ...

Added: April 2, 2020

On the functor of Arakawa, Suzuki and Tsuchiya

Khoroshkin S. M., Nazarov M., , in: Advanced Studies in Pure Mathematics (т.76 Representations Theory, Special Functions and Painleve Equations - RIMS 2015)Vol. 76.: Tokyo: Mathematical Society of Japan, 2018. Ch. 8 P. 275–302.

Arakawa, Suzuki and Tsuchiya defined a correspondence between certain modules of the trigonometric Cherednik algebra CN depending on a parameter κ∈C, and certain modules of the affine Lie algebra slˆm of level κ−m. We give a detailed proof of this correspondence by working with the affine Lie algebra glˆm alongside of slˆm. We also relate ...

Added: October 8, 2019

Conjecture on theta-blocks of order 1

Valery Gritsenko, Wang H., Russian Mathematical Surveys 2017 Vol. 72 No. 5 P. 968–970

In this paper we prove the indicated conjecture in the last case of known infinite series of theta-blocks of weight 2. ...

Added: January 29, 2018

Гипотеза о тэта-блоках первого порядка

Gritsenko V., Ванг Х., Успехи математических наук 2017 Т. 72 № 5 С. 191–192

In this paper we prove the conjecture above in the last case of known theta-blocks of weight 2. This gives a new intereting series of Borcherds products of weight 2. ...

Added: October 11, 2017

On Lagrangian spheres in the flag variety F 3

Tyurin N. A., Mathematical notes 2015 Vol. 98 No. 1-2 P. 348–351

Added: October 7, 2015

Weyl's Formula as the Brion Theorem for Gelfand-Tsetlin Polytopes

Makhlin I., / Series math "arxiv.org". 2014.

We exploit the idea that the character of an irreducible finite dimensional gln-module is the sum of certain exponents of integer points in a Gelfand-Tsetlin polytope and can thus be calculated via Brion's theorem. In order to show how the result of such a calculation matches Weyl's character formula we prove some interesting combinatorial traits ...

Added: August 7, 2015

Characters of Feigin-Stoyanovsky subspaces and Brion's theorem

Makhlin I., Functional Analysis and Its Applications 2015 Vol. 49 No. 1 P. 15–24

We give an alternative proof of the main result of [1]; the proof relies on Brion’s theorem about convex polyhedra. The result itself can be viewed as a formula for the character of the Feigin-Stoyanovsky subspace of an integrable irreducible representation of the affine Lie algebra widehatsln(C). Our approach is to assign integer points of a ...

Added: August 7, 2015

24 faces of the Borcherds modular form Phi_{12}

Gritsenko V., / Series math "arxiv.org". 2012. No. 6503.

The fake monster Lie algebra is determined by the Borcherds function Phi_{12} which is the reflective modular form of the minimal possible weight with respect to O(II_{2,26}). We prove that the first non-zero Fourier-Jacobi coefficient of Phi_{12} in any of 23 Niemeier cusps is equal to the Weyl-Kac denominator function of the affine Lie algebra ...

Added: March 3, 2015

Degenerate affine Grassmannians and loop quivers

Feigin E., Finkelberg M. V., Reineke M., / Series math "arxiv.org". 2014. No. 1410.0777.

We study the connection between the affine degenerate Grassmannians in type $A$, quiver Grassmannians for one vertex loop quivers and affine Schubert varieties. We give an explicit description of the degenerate affine Grassmannian of type $GL_n$ and identify it with semi-infinite orbit closure of type $A_{2n-1}$. We show that principal quiver Grassmannians for the one vertex loop ...

Added: October 6, 2014

Finite-dimensional subalgebras in polynomial Lie algebras of rank one.

Arzhantsev I., Makedonskii E. A., Petravchuk A. P., Ukrainian Mathematical Journal 2011 Vol. 63 No. 5 P. 708–712

Added: July 10, 2014

Коммутативные вертексные алгебры и их вырождения

Feigin B. L., Функциональный анализ и его приложения 2014 № 3

We study commutative vertex operator algebras. These algebras are isomorphic to certain subalgebras in Kac-Moody vertex operator algebras. We describe systems of relations and degenerations to quadratic algebras. Our approach leads to the fermionic formulas for characters. ...

Added: April 14, 2014

Projections of orbital measures, Gelfand-Tsetlin polytopes, and splines

Olshanski G., Journal of Lie Theory 2013 Vol. 23 No. 4 P. 1011–1022

The unitary group U(N) acts by conjugations on the space H(N) of NxN Hermitian matrices, and every orbit of this action carries a unique invariant probability measure called an orbital measure. Consider the projection of the space H(N) onto the real line assigning to an Hermitian matrix its (1,1)-entry. Under this projection, the density of ...

Added: November 24, 2013