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

Cornell University
,
2015.

Feigin B. L., Makhlin I.

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.

Makhlin I., / Cornell University. 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

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

Makhlin I., Selecta Mathematica, New Series 2015

We exploit the idea that the character of an irreducible finite dimensional $\mathfrak{gl}_n$-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: September 29, 2014

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

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

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

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

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

Dumanski I., Feigin E., Finkelberg M. V., / Cornell University. 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

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

Added: October 7, 2015

Feigin E., Finkelberg M. V., Reineke M., / Cornell University. 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

Arzhantsev I., Makedonskii E. A., Petravchuk A. P., Украинский математический журнал 2011 Vol. 63 No. 5 P. 708-712

Added: July 10, 2014

Gritsenko V., / Cornell University. 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

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