arXiv subjects

Khanh Nguyen Duc

Publications and source records attributed to Khanh Nguyen Duc.

7 recordsLinked to original sources

arXiv2026

Branching rules for winding subalgebras of the affine Kac--Moody algebras $A^{(1)}_1$ and $A^{(2)}_2$

We study branching problems for affine Kac--Moody algebras. Unlike the finite-dimensional case, an affine Kac--Moody algebra may contain proper subalgebras isomorphic to itself, such as winding subalgebras obtained by rescaling the loop parameter. We investigate the restriction of integrable highest-weight representations to such subalgebras. The restriction remains integrable and decomposes into irreducible components with finite multiplicities, encoded by pairs of highest weights. We show that this set is closed under addition, extending a result of Brion and Knop to the affine setting. We also give a partial description of this set and provide explicit results for types $A^{(1)}_1$ and $A^{(2)}_2$.

math.RT↗

arXiv2026

Steenrod Lengths and a Problem of Vakil

We give an explicit combinatorial description of the function $f(n)$ governing the Steenrod length of real projective spaces $\mathbb{RP}^n$. This function arises in stable homotopy theory through the action of Steenrod squares on mod-$2$ cohomology and is closely related to the ghost length, which measures the minimal number of spheres required to construct a space up to homotopy. Building on the directed graphs $T_n$ introduced by Vakil to encode degree constraints for Steenrod operations, we interpret $f(n)$ as the length of the longest directed path starting at $n$. Using this framework, we resolve a question posed by Vakil by deriving concrete combinatorial formulas for $f(n)$ in terms of binary classes and a distinguished family of integers, which we call Vakil numbers.

math.RT↗

arXiv2025

Branching rule on winding subalgebras of affine Kac-Moody algebras

In this paper, by using the Lakshmibai-Seshadri paths, we give the branching rule for representations of affine Kac-Moody algebras to their winding subalgebras. As a corollary, we can describe branching multiplicities in the language of paths. An analog of Steinberg's formula for branching multiplicities is also given.

math.RT↗

arXiv2024

A Murnaghan-Nakayama rule for Grothendieck polynomials of Grassmannian type

We consider the Grothendieck polynomials appearing in the K-theory of Grassmannians, which are analogs of Schur polynomials. This paper aims to establish a version of the Murnaghan-Nakayama rule for Grothendieck polynomials of the Grassmannian type. This rule allows us to express the product of a Grothendieck polynomial with a power sum symmetric polynomial into a linear combination of other Grothendieck polynomials.

math.CO↗

arXiv2023

A generalization of the Murnaghan-Nakayama rule for $K$-$k$-Schur and $k$-Schur functions

The $K$-$k$-Schur functions and $k$-Schur functions appeared in the study of $K$-theoretic and affine Schubert Calculus as polynomial representatives of Schubert classes. In this paper, we introduce a new family of symmetric functions $\mathcal{F}_λ^{(k)}$, that generalizes the constructions via the Pieri rule of $K$-$k$-Schur functions and $ k$-Schur functions. Then we obtain the Murnaghan-Nakayama rule for the generalized functions. The rule is described explicitly in the cases of $K$-$k$-Schur functions and $k$-Schur functions, with concrete descriptions and algorithms for coefficients. Our work recovers the result of Bandlow, Schilling, and Zabrocki for $k$-Schur functions, and explains it as a degeneration of the rule for $K$-$k$-Schur functions. In particular, many other special cases and connections promise to be detailed in the future.

math.RT↗

arXiv2023

Newton polytope of good symmetric polynomials

We introduce a general class of symmetric polynomials that have saturated Newton polytope and their Newton polytope has integer decomposition property. The class covers numerous previously studied symmetric polynomials.

math.CO↗

arXiv2022

On the Shifted Littlewood-Richardson Coefficients and Littlewood-Richardson Coefficients

We give a new interpretation of the shifted Littlewood-Richardson coefficients $f_{λμ}^ν$ ($λ,μ,ν$ are strict partitions). The coefficients $g_{λμ}$ which appear in the decomposition of Schur $Q$-function $Q_λ$ into the sum of Schur functions $Q_λ= 2^{l(λ)}\sum_μg_{λμ}s_μ$ can be considered as a special case of $f_{λμ}^ν$ (here $λ$ is a strict partition of length $l(λ)$). We also give another description for $g_{λμ}$ as the cardinal of a subset of a set that counts Littlewood-Richardson coefficients $c_{μ^tμ}^{\tildeλ}$. This new point of view allows us to establish connections between $g_{λμ}$ and $c_{μ^t μ}^{\tildeλ}$. More precisely, we prove that $g_{λμ}=g_{λμ^t}$, and $g_{λμ} \leq c_{μ^tμ}^{\tildeλ}$. We conjecture that $g_{λμ}^2 \leq c^{\tildeλ}_{μ^tμ}$ and formulate some conjectures on our combinatorial models which would imply this inequality if it is valid.

math.RT↗