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Ajeeth Gunna

Publications and source records attributed to Ajeeth Gunna.

6 recordsLinked to original sources
arXiv2026

Inhomogeneous $q$-Whittaker polynomials II: ring theorem and positive specializations

We study inhomogeneous $q$-Whittaker polynomials which extend both $q$-Whittaker and stable Grothendieck polynomials. We prove that inhomogeneous $q$-Whittaker polynomials (in countably many variables) form a basis of certain commutative ring extending the ring of symmetric functions to a subring of its completion. We then describe positive specializations of that ring and relate them with a subset of Macdonald-positive specializations of the ring of symmetric functions. We also show some related probability distributions obtained from positive specializations of inhomogeneous $q$-Whittaker polynomials.

math.CO↗
arXiv2025

Inhomogeneous $q$-Whittaker Polynomials I: Duality and Expansions

We introduce a new family of symmetric polynomials $\mathfrak{G}^{(\mathbf{u},\mathbf{v})}_{\lambda}$ arising from exactly solvable lattice models associated with the quantised loop algebra $\mathcal{U}_{q}(\mathfrak{sl}_{2}[z^\pm])$. The polynomials $\mathfrak{G}^{(\mathbf{u},\mathbf{v})}_{\lambda}$ unify $q$-Whittaker polynomials, inhomogeneous $q$-Whittaker polynomials, Grothendieck polynomials and their duals. Using Yang--Baxter equation, we derive Cauchy identities and combinatorial formulas for the transition coefficients.

math.CO↗
arXiv2025

Structure constants for spin Hall--Littlewood functions

We provide a combinatorial formula for the structure constants of spin Hall--Littlewood functions. This is achieved by representing these functions and the structure constants as the partition function of a lattice model and applying the underlying Yang--Baxter equation. Our combinatorial expression is in terms of generalised honeycombs; the latter were introduced by Knutson and Tao for ordinary Littlewood--Richardson coefficients and applied to the computation of Hall polynomials by Zinn--Justin.

math.CO↗
arXiv2023

Shuffle algebras, lattice paths and Macdonald functions

We consider partition functions on the $N\times N$ square lattice with the local Boltzmann weights given by the $R$-matrix of the $U_{t}(\widehat{sl}(n+1|m))$ quantum algebra. We identify boundary states such that the square lattice can be viewed on a conic surface. The partition function $Z_N$ on this lattice computes the weighted sum over all possible closed coloured lattice paths with $n+m$ different colours: $n$ ``bosonic'' colours and $m$ ``fermionic'' colours. Each bosonic (fermionic) path of colour $i$ contributes a factor of $z_i$ ($w_i$) to the weight of the configuration. We show the following. (i) $Z_N$ is a symmetric function in the spectral parameters $x_1\dots x_N$ and generates basis elements of the commutative trigonometric Feigin--Odesskii shuffle algebra. The generating function of $Z_N$ admits a shuffle-exponential formula analogous to the Macdonald Cauchy kernel. (ii) $Z_N$ is a symmetric function in two alphabets $(z_1\dots z_n)$ and $(w_1\dots w_m)$. When $x_1\dots x_N$ are set to be equal to the box content of a skew Young diagram $\mu/\nu$ with $N$ boxes the partition function $Z_N$ reproduces the skew Macdonald function $P_{\mu/\nu}\left[w-z\right]$.

math-ph↗
arXiv2022

Integrable systems and crystals for edge labeled tableaux

We introduce the edge Schur functions $E^λ$ that are defined as a generating series over edge labeled tableaux. We formulate $E^λ$ as the partition function for a solvable lattice model, which we use to show they are symmetric polynomials and derive a Cauchy-type identity with factorial Schur polynomials. Finally, we give a crystal structure on edge labeled tableau to give a positive Schur polynomial expansion of $E^λ$ and show it intertwines with an uncrowding algorithm.

math.CO↗