arXiv subjects

G. Krishna Teja

Publications and source records attributed to G. Krishna Teja.

7 recordsLinked to original sources

arXiv2026

Characters of modules over negative rank-2 Borcherds-Kac-Moody Lie algebras

Let $\mathfrak{g}=\mathfrak{g}(A)$ be the Borcherds-Kac-Moody Lie algebra (BKM LA) for a BKM Cartan matrix $A$ that is filled by negative integers. Fix a Cartan subalgebra $\mathfrak{h}$ of $\mathrm{g}$ and the classical cone of dominant integral weights $P^+\subset \mathfrak{h}^*$. The non-integrable simple highest weight $\mathfrak{g}$-modules $L(μ)$'s widely studied were those by Naito ([Trans. Amer. Soc., 1995]), for $μ$'s dot-linked to $P^+$-translates of sums $- \sum_{j\in J}α_j$ of mutually orthogonal and imaginary simple roots $α_j$'s. Recently, we computed weights of all highest weight $\mathfrak{g}$-modules $V$'s (over all BKM LA's), and character of $L(ρ)$ for Weyl vector $ρ$. These needed a family of ``integrable'' $L(μ)$'s for $μ$'s inside our novel signed-dominant-integral cone $P^{\pm}$ (which generalizes $P^+$). Pairings $μ(α_i^{\vee})\leq 0$ for $μ\in P^{\pm}$ are multiples of $\frac{A_{ii}}{2}$ for all $i$. Nevertheless, $L(μ)$ contains ``Chevalley-Serre relations'' $f_i^{\frac{2}{A_{ii}}{μ(α_i^{\vee})}+1}L(μ)_μ=0$, which seem to be previously unstudied and even in Naito's works. This paper initiates in rank-2, the study of module structures and maximal vectors (or Verma embeddings) in Verma covers $M(μ)$'s of $L(μ)$'s for $μ\in P^{\pm}$. Our goal in this is to explore in weight spaces of those Vermas, the strictness, or else a uniform equality, of lower bounds by Kac and Kazhdan ([Adv. Math., 1979]) for count of linearly independent maximal vectors. We obtain presentations and characters of all $V$'s when Kac-Kazhdan equation has unique solution in the interior of root-cone. This builds on results of Kac and Kazhdan in crucial unique solution case.

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arXiv2026

Weights and characters of highest weight modules

Let $\mathfrak{g}=\mathfrak{g}(A)$ be any Borcherds-Kac-Moody $\mathbb{C}$-Lie algebra (BKM LA) for BKM-Cartan matrix $A$, with Cartan subalgebra $\mathfrak{h}$. Let $V$ denote a highest weight $\mathfrak{g}$-module, with top weight $λ\in \mathfrak{h}^*$ (not necessarily in the domninant integral cone $P^+$). The non-integrable simples $V= L(λ)$ by Naito ([Trans. Amer. Soc., 1995]) are widely studied beyond integrable simple $L(ν)s,\ ν\in P^+$. We introduce and study: 1) A weight cone $P^{\pm}=\big\{μ\in \mathfrak{h}^*\ \big|\ μ(α_i^{\vee})\in \frac{A_{ii}}{2}\mathbb{Z}_{\geq 0}\text{ for all simple co-roots }α_i^{\vee}\big\}$; note Weyl vector $ρ\in P^{\pm}\setminus P^+$. 2) The resulting (novel) non-integrable simple $L(λ)s, \ λ\in P^{\pm}\setminus P^{+}$; their Chevalley-Serre (CS) type relations (which are, in fact, complementary to those of integrable $L(ν)$s); 3) Higher length CS type relations in any highest weight module under the name ``holes". Using these, we obtain explicitly and uniformly, (notably) Weyl-orbit typed formulas for weight-sets of: all simples $L(λ)$s ($\forall$ $λ\in \mathfrak{h}^*$) and all quotients of parabolic Verma modules along imaginary directions. This generalizes and extends in one stroke, such formulas over Kac-Moody (KM) $\mathfrak{g}$, of all $L(λ)$ by Khare ([Trans. Amer. Math. Soc. 2017]), and Dhillon and Khare ([Adv. Math., 2017], and also of all $V$ by Khare and Teja recently; which used parabolic and higher order Verma modules. We obtain Weyl-Kac-Borcherds type character formulas for $L(λ) \text{ for } λ\in P^{\pm}$, over negative rank-2 $\mathfrak{g}$'s; by exploring Verma module embeddings. We obtain character of every highest weight module $V$ for $λ=ρ$ in negative $A$-type cases.

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arXiv2026

Characterization of Erdös matrices by their zero entries

An Erdös matrix $E$ is a bistochastic matrix whose sum of squares of entries (Frobenius norm squared) equals its maxtrace (maximum of all the $σ$-traces for permutations $σ$'s). We characterize all Erdös $E$ by the patterns of their zero entries; showing that each such skeleton has at most one $E$. We present an algorithm to find all $n\times n$ Erdös matrices, which finds them up to $n\leqslant 5$ quickly and also size $n=6$. We further show some presently known RCDS matrices to be Erdös.

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arXiv2025

A weight-formula for all highest weight modules, and a higher order parabolic category $\mathcal{O}$

Let $\mathfrak{g}$ be a complex Kac-Moody algebra, with Cartan subalgebra $\mathfrak{h}$. Also fix a weight $λ\in\mathfrak{h}^*$. For $M(λ)\twoheadrightarrow V$ an arbitrary highest weight $\mathfrak{g}$-module, we provide a cancellation-free, non-recursive formula for the weights of $V$. This is novel even in finite type, and is obtained from $λ$ and a collection $\mathcal{H}=\mathcal{H}_V$ of independent sets in the Dynkin diagram of $\mathfrak{g}$ that are associated to $V$. Our proofs use and reveal a finite family (for each $λ$) of "higher order Verma modules" $\mathbb{M}(λ,\mathcal{H})$ - these are all of the universal modules for weight-considerations. They (i) generalize and subsume parabolic Verma modules $M(λ,J)$, and (ii) have pairwise distinct weight-sets, which exhaust the weight-sets of all modules $M(λ)\twoheadrightarrow V$. As an application, we explain the sense in which the modules $M(λ)$ of Verma and $M(λ,J_V)$ of Lepowsky are respectively the zeroth and first order upper-approximations of every $V$, and continue to higher order upper-approximations $\mathbb{M}_k(λ,\mathcal{H}_V)$ (and to lower-approximations). We determine every $k$th order integrability of $V$. We then introduce the category $\mathcal{O}^\mathcal{H}\subset\mathcal{O}$, which is a higher order parabolic analogue that contains the higher order Verma modules $\mathbb{M}(λ,\mathcal{H})$. We show that $\mathcal{O}^\mathcal{H}$ has enough projectives, and also initiate the study of BGG reciprocity, by proving it for all $\mathcal{O}^\mathcal{H}$ over $\mathfrak{g}=\mathfrak{sl}_2^{\oplus n}$. Finally, we provide a BGG resolution for our universal modules $\mathbb{M}(λ,\mathcal{H})$ in certain cases including all rank-3 $\mathfrak{g}$; this yields their Weyl-type character formulas, with the actions of parabolic Weyl semigroups.

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arXiv2024

Minkowski difference weight formulas

Fix any complex Kac-Moody Lie algebra $\mathfrak{g}$, and Cartan subalgebra $\mathfrak{h}\subset \mathfrak{g}$. We study arbitrary highest weight $\mathfrak{g}$-modules $V$ (with any highest weight $λ\in \mathfrak{h}^*$, and let $L(λ)$ be the corresponding simple highest weight $\mathfrak{g}$-module), and write their weight-sets $\mathrm{wt} V$. This is based on and generalizes the Minkowski decompositions for all $\mathrm{wt} L(λ)$ and hulls $\mathrm{conv}_{\mathbb{R}}(\mathrm{wt} V)$, of Khare [J. Algebra. 2016 & Trans. Amer. Math. Soc. 2017] and Dhillon-Khare [Adv. Math. 2017 & J. Algebra. 2022]. Those works need a freeness property of the Dynkin graph nodes of integrability $J_λ$ of $L(λ)$: $\mathrm{wt} L(λ)\ -$ any sum of simple roots over $J_λ^c$ are all weights of $L(λ)$. We generalize it for all $V$, by introducing nodes $J_V$ that record all the lost 1-dim. weights in $V$. We show three applications (seemingly novel) for all $\big(\mathfrak{g}, λ, V\big)$ of our $J_V^c$-freeness: 1) Minkowski decompositions of all $\mathrm{wt} V$, subsuming those above for simples. 1$'$) Characterization of these formulas. 1$''$) For these, we solve the inverse problem of determining all $V$ with fixing $\mathrm{wt} V \ =$ weight-set of a Verma, parabolic Verma and $L(λ)$ $\forall$ $λ$. 2) At module level (by raising operators' actions), construction of weight vectors along $J_V^c$-directions. 3) Lower bounds on the multiplicities of such weights, in all $V$.

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arXiv2022

Weak faces and a formula for weights of highest weight modules, via parabolic partial sum property for roots

Let $\mathfrak{g}$ be a finite or an affine type Lie algebra over $\mathbb{C}$ with root system $Δ$. We show a parabolic generalization of the partial sum property for $Δ$, which we term the parabolic partial sum property. It allows any root $β$ involving (any) fixed subset $S$ of simple roots, to be written as an ordered sum of roots, each involving exactly one simple root from $S$, with each partial sum also being a root. We show three applications of this property to weights of highest weight $\mathfrak{g}$-modules: (1)~We provide a minimal description for the weights of all non-integrable simple highest weight $\mathfrak{g}$-modules, refining the weight formulas shown by Khare [J. Algebra} 2016] and Dhillon-Khare [Adv. Math. 2017]. (2)~We provide a Minkowski difference formula for the weights of an arbitrary highest weight $\mathfrak{g}$-module. (3)~We completely classify and show the equivalence of two combinatorial subsets - weak faces and 212-closed subsets - of the weights of all highest weight $\mathfrak{g}$-modules. These two subsets were introduced and studied by Chari-Greenstein [Adv. Math. 2009], with applications to Lie theory including character formulas. We also show ($3'$) a similar equivalence for root systems.

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arXiv2021

Weak faces of highest weight modules and root systems

Chari and Greenstein [Adv. Math. 2009] introduced combinatorial subsets of the roots of a finite-dimensional simple Lie algebra $\mathfrak{g}$ which were important in studying Kirillov-Reshetikhin modules over $U_q(\widehat{\mathfrak{g}})$ and their specializations. Later, Khare [J. Algebra. 2016] studied these subsets for many highest weight $\mathfrak{g}$-modules (in finite type), under the name of weak-$\mathbb{A}$-faces (for a subgroup $\mathbb{A}$ of $(\mathbb{R},+)$), and more generally, $(\{2\};\{1,2\})$-closed subsets. These notions extend and unify the faces of Weyl polytopes as well as the above combinatorial subsets. In this paper, we consider these 'discrete' notions for all Kac-Moody algebras $\mathfrak{g}$, in four distinguished settings: (a) the weights of an arbitrary highest weight $\mathfrak{g}$-module $V$; (b) the convex hull of the weights of $V$; (c) the weights of the adjoint representation; (d) the roots of $\mathfrak{g}$. For (a) (resp., (b)) for all highest weight $\mathfrak{g}$-modules $V$, we show that the weak-$\mathbb{A}$-faces and $(\{2\};\{1,2\})$-closed subsets agree, and equal the sets of weights on exposed faces (resp., equal the exposed faces) of the convex hull of weights conv$_{\mathbb{R}}$ wt $V$. This completes the partial progress of Khare in finite type, and is novel in infinite type. Our proofs are type-free and self-contained. For (c), (d) involving the root system, we similarly achieve complete classifications. For all Kac-Moody $\mathfrak{g}$ - interestingly, other than $\mathfrak{sl}_3, \widehat{\mathfrak{sl}_3}$ - we show the weak-$\mathbb{A}$-faces and $(\{2\};\{1,2\})$-closed subsets agree, and equal Weyl group translates of the sets of weights in certain 'standard faces' (which also holds for highest weight modules). This was proved by Chari and her coauthors for root systems in finite type, but is novel for other types.

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