An official website of the United States government
We introduce and study an inhomogeneous generalization of the spin -Whittaker polynomials from [15]. These are symmetric polynomials, and we prove a branching rule, skew dual and non-dual Cauchy identities, and an integral representation for them. Our main tool is a novel family of deformed Yang–Baxter equations.
more »
« less
This content will become publicly available on July 25, 2026
Symmetries of periodic and free boundary q-Whittaker measures
We show that the periodic and free boundary q-Whittaker measures, two models of random partitions, exhibit remarkable distributional symmetries. Equivalently, we derive new identities for skew q-Whittaker functions related to bounded Cauchy and Littlewood identities. These extend identities found by Imamura, Mucciconi, and Sasamoto, and in particular give new proofs of these identities.
more »
« less
- Award ID(s):
- 2451487
- PAR ID:
- 10653717
- Publisher / Repository:
- Séminaire Lotharingien de Combinatoire
- Date Published:
- Journal Name:
- Collection Art contemporain
- ISSN:
- 1286-4889
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Alexeev, A.; Frenkel, E.; Rosso, M.; Webster, B.; Yakimov, M. (Ed.)We propose solutions of the quantum Q-systems of types BN,CN,DN in terms of q-difference operators, generalizing our previous construction for the Q- system of type A. The difference operators are interpreted as q-Whittaker limits of discrete time evolutions of Macdonald-van Diejen type operators. We conjecture that these new operators act as raising and lowering operators for q-Whittaker functions, which are special cases of graded characters of fusion products of KR- modules.more » « less
-
A bstract We investigate the underlying quantum group symmetry of 2d Liouville and dilaton gravity models, both consolidating known results and extending them to the cases with $$ \mathcal{N} $$ N = 1 supersymmetry. We first calculate the mixed parabolic representation matrix element (or Whittaker function) of U q ( $$ \mathfrak{sl} $$ sl (2 , ℝ)) and review its applications to Liouville gravity. We then derive the corresponding matrix element for U q ( $$ \mathfrak{osp} $$ osp (1 | 2 , ℝ)) and apply it to explain structural features of $$ \mathcal{N} $$ N = 1 Liouville supergravity. We show that this matrix element has the following properties: (1) its q → 1 limit is the classical OSp + (1 | 2 , ℝ) Whittaker function, (2) it yields the Plancherel measure as the density of black hole states in $$ \mathcal{N} $$ N = 1 Liouville supergravity, and (3) it leads to 3 j -symbols that match with the coupling of boundary vertex operators to the gravitational states as appropriate for $$ \mathcal{N} $$ N = 1 Liouville supergravity. This object should likewise be of interest in the context of integrability of supersymmetric relativistic Toda chains. We furthermore relate Liouville (super)gravity to dilaton (super)gravity with a hyperbolic sine (pre)potential. We do so by showing that the quantization of the target space Poisson structure in the (graded) Poisson sigma model description leads directly to the quantum group U q ( $$ \mathfrak{sl} $$ sl (2 , ℝ)) or the quantum supergroup U q ( $$ \mathfrak{osp} $$ osp (1 | 2 , ℝ)).more » « less
-
Abstract We construct a family of solvable lattice models whose partition functions include ‐adic Whittaker functions for general linear groups from two very different sources: from Iwahori‐fixed vectors and from metaplectic covers. Interpolating between them by Drinfeld twisting, we uncover unexpected relationships between Iwahori and metaplectic Whittaker functions. This leads to new Demazure operator recurrence relations for spherical metaplectic Whittaker functions. In prior work of the authors it was shown that the row transfer matrices of certain lattice models for spherical metaplectic Whittaker functions could be represented as ‘half‐vertex operators’ operating on the ‐Fock space of Kashiwara, Miwa and Stern. In this paper the same is shown for all the members of this more general family of lattice models including the one representing Iwahori Whittaker functions.more » « less
-
Employing bijectivization of summation identities, we introduce local stochastic moves based on the Yang–Baxter equation for $$U_{q}(\widehat{\mathfrak{sl}_{2}})$$ . Combining these moves leads to a new object which we call the spin Hall–Littlewood Yang–Baxter field—a probability distribution on two-dimensional arrays of particle configurations on the discrete line. We identify joint distributions along down-right paths in the Yang–Baxter field with spin Hall–Littlewood processes, a generalization of Schur processes. We consider various degenerations of the Yang–Baxter field leading to new dynamic versions of the stochastic six-vertex model and of the Asymmetric Simple Exclusion Process.more » « less
