arXiv subjects

Deepanshu Prasad

Publications and source records attributed to Deepanshu Prasad.

4 recordsLinked to original sources

arXiv2026

Counting 3-way contingency tables via quiver semi-invariants

Let $\mathbf{T}_{\mathbf{a},\mathbf{b}}$ be the number of $3$-way contingency tables of size $m \times n \times p$ with two of its three plane-sum margins fixed by $\mathbf{a}=(a_1, \ldots, a_m) \in \mathbb{N}^m$ and $\mathbf{b}=(b_1, \ldots, b_n) \in \mathbb{N}^n$. When $p=1$, this is the number of $m \times n$ non-negative integer matrices whose row and column sums are fixed by $\mathbf{a}$ and $\mathbf{b}$. In this paper, we study the numbers $\mathbf{T}_{\mathbf{a},\mathbf{b}}$ through the lens of quiver invariant theory. Let $\mathcal{Q}^{p}_{m,n}$ be the $p$-complete bipartite quiver with $m$ source vertices, $n$ sink vertices, and $p$ arrows from each source to each sink. Let $\mathbf{1}$ denote the dimension vector of $\mathcal{Q}^{p}_{m,n}$ that takes value $1$ at every vertex of $\mathcal{Q}^{p}_{m,n}$, and let $\theta_{\mathbf{a}, \mathbf{b}}$ denote the integral weight that assigns $a_i$ to the $i^{th}$ source vertex and $-b_j$ to the $j^{th}$ sink vertex of $\mathcal{Q}^{p}_{m,n}$. We begin by realizing $\mathbf{T}_{\mathbf{a},\mathbf{b}}$ as the dimension of the space of semi-invariants associated to $(\mathcal{Q}^{p}_{m,n}, \mathbf{1}, \theta_{\mathbf{a}, \mathbf{b}})$. Using this connection and methods from quiver invariant theory, we show that $\mathbf{T}_{\mathbf{a},\mathbf{b}}$ is a parabolic Kostka coefficient. In the case $p=1$, this recovers the formula for the number of the $m \times n$ contingency tables with row and column sums fixed by $\mathbf{a}$ and $\mathbf{b}$, which in the classical $2$-way setting can also be obtained via the Robinson-Schensted-Knuth correspondence.

math.CO↗

arXiv2026

Growth of infinite frieze patterns of affine type

We analyse the growth coefficients of infinite frieze patterns arising from cluster algebras using cluster modular groups and cluster categories. For a fixed cluster category of affine type, we prove that the collection of infinite frieze patterns given by both the homogeneous and non-homogeneous stable tubes all have the same growth coefficients. We also derive and verify an explicit formula for the $k$-th growth coefficient, expressed directly in terms of data from homogeneous tubes, or, alternatively, from appropriate elements of the corresponding cluster algebra.

math.CO↗

arXiv2024

Galois Coverings, $\tau$-Rigidity and Mutations

For an algebraically closed field $\mathbb{K}$, we consider a Galois $G$-covering $\mathcal{B} \to \mathcal{A}$ between locally bounded $\mathbb{K}$-categories given by bound quivers, where $G$ is torsion-free and acts freely on the objects of $\mathcal{B}$. We define the notion of $(G,\tau_{\mathcal{B}})$-rigid subcategory and of support $(G,\tau_{\mathcal{B}})$-tilting pairs over $\mathcal{B}$-$\rm mod$. These are the analogues of the similar concepts in the context of a finite-dimensional algebra, where we additionally require that the subcategory be $G$-equivariant. When $\mathcal{A}$ is a finite-dimensional algebra, we show that the corresponding push-down functor $\mathcal{F}_{\lambda}: \mathcal{B}$-$\rm mod$ $\to \mathcal{A}$-$\rm mod$ sends $(G,\tau_{\mathcal{B}})$-rigid subcategories (respectively support $(G,\tau_{\mathcal{B}})$-tilting pairs) to $\tau_{\mathcal{A}}$-rigid modules (respectively support $\tau_{\mathcal{A}}$-tilting pairs). We further show that there is a notion of mutation for support $(G,\tau_{\mathcal{B}})$-tilting pairs over $\mathcal{B}$-$\rm mod$. Mutations of support $\tau_\mathcal{A}$-tilting pairs and of support $(G,\tau_\mathcal{B})$-tilting pairs commute with the push-down functor. We derive some consequences of this, and in particular, we derive a $\tau$-tilting analogue of the result of P. Gabriel that locally representation-finiteness is preserved under coverings. Finally, we prove that when the Galois group $G$ is finitely generated free, any rigid $\mathcal{A}$-module (and in particular $\tau_\mathcal{A}$-rigid $\mathcal{A}$-modules) lies in the essential image of the push-down functor.

math.RT↗

arXiv2023

Semi-Invariant Rings: UFD and Codimension One Orbits

Let $A$ be a finite dimensional associative $\mathbb{K}$-algebra over an algebraically closed field $\mathbb{K}$ of characteristic zero. To $A$, we can associate its basic form that is given by a quiver $Q = (Q_0, Q_1)$ with an admissible ideal $R$. For a dimension vector $β$, we consider an irreducible component $\mathcal{C}$ of the module variety of $β$-dimensional representations of $A$. The reductive group ${\rm GL}_β(\mathbb{K}):= \prod_{i \in Q_0}{\rm GL}_{β_i}(\mathbb{K})$ acts on $\mathcal{C}$ by change of basis, and has a unique closed orbit. We consider the corresponding ring of semi-invariants ${\rm SI}(Q, \mathcal{C})$. We prove that if $\mathcal{C}$ is factorial and has maximal orbits of codimension one, then ${\rm SI}(Q, \mathcal{C})$ is a complete intersection and is not multiplicity free. If $\mathcal{C}$ is not factorial, then this conclusion does not necessarily hold. We present examples showing that the codimension of the complete intersection can be arbitrarily large. Finally, we interpret our results in the case of hereditary algebras.

math.RT↗