arXiv subjects

Santosha Pattanayak

Publications and source records attributed to Santosha Pattanayak.

11 recordsLinked to original sources

arXiv2026

Fano Generalized Bott-Samelson Varieties

The Bott-Samelson varieties provide natural desingularizations of Schubert varieties, and their generalizations, called generalized Bott-Samelson varieties, were constructed by Nicolas Perrin as towers of locally trivial fibrations with fibers isomorphic to Schubert varieties. In this article, we give a complete characterization of Fano and weak Fano generalized Bott-Samelson varieties. As a consequence, we recover the corresponding results for ($G$-)Bott-Samelson varieties and minuscule generalized Bott-Samelson varieties.

math.AG↗

arXiv2026

Geometric Invariant Theory of Peterson Varieties

We study the GIT quotients of the Peterson variety $\mathrm{Pet}_n\subset \mathrm{GL}(n,\mathbb C)/B$ under a one-parameter subgroup $λ:\mathbb G_m \to T$ with respect to the linearization $\mathcal L(χ)$ given by a regular dominant character $χ$ in the root lattice. Using the Richardson stratification, we describe the semistable and stable loci explicitly in terms of subsets of simple roots. This determines the GIT chamber decomposition and the corresponding wall-crossing morphisms. In the deep chamber, the quotient is shown to be isomorphic to the weighted projective space $\mathbb P(1,2,\ldots,n-1)$. We obtain a complete chamber-theoretic characterization of normality and describe how normality varies with the choice of linearization. We also prove that the quotient is smooth if and only if $n\le3$, independently of the regular dominant linearization. These results describe how the singular geometry of the Peterson variety is reflected in the variation of its GIT quotients.

math.AG↗

arXiv2026

A Newton Identity and Finite-Rank Reconstruction for the Queer Lie Superalgebra

We establish a Newton-type identity for the queer Lie superalgebra $\mathfrak q_N$, relating Sergeev's odd cyclic central elements to Nazarov's one-row Capelli elements. The identity is obtained by comparing Ivanov's generating function for factorial Schur $Q$-functions with the queer Perelomov-Popov product of Grigoryev and Nazarov. Its coefficient expansion yields a triangular change of generators between the odd cyclic and odd one-row families. In particular, the odd one-row Capelli elements generate the center, while the even one-row elements are redundant. In fixed rank, we derive determinantal relations and a generic reconstruction theorem. The basic cyclic Hankel determinant is identified with a resultant and factored into the failure-of-strong-typicality and shifted-resonance factors. After localization at this determinant, the center is generated by the first $2N$ odd cyclic elements; consequently, generic central characters are determined by their values on these elements.

math.CO↗

arXiv2026

Uniqueness of Branching through regular unipotent elements

Let $\mathrm G$ be a complex simple algebraic group and let $\mathrm G_0\subset \mathrm G$ be a closed connected subgroup containing a regular unipotent element of $\mathrm G$, with semisimple rank at least $2$. Using Dynkin's classification, we prove that the restriction of an irreducible finite-dimensional representation of $\mathrm G$ to $\mathrm G_0$ determines the representation up to an outer automorphism of $\mathrm G$ preserving $\mathrm G_0$. We extend this method to the diagonal embedding $\mathrm G_0\hookrightarrow \mathrm G\times \mathrm G$ for the specific pairs $(\mathrm{SO}_{2k}(\mathbb C) \times\mathrm{SO}_{2k}(\mathbb C),\,\mathrm{SO}_{2k-1}(\mathbb C))$, $(E_6\times E_6,\,F_4)$ and $(Spin_8(\mathbb C) \times Spin_8(\mathbb C), G_2)$ and show that uniqueness continues to hold. Finally, we give examples showing that, in the diagonal setting, restriction to the principal $\mathrm{SL}_2(\mathbb C)$ alone is not sufficient to establish uniqueness.

math.RT↗

arXiv2025

On the GIT quotient of Grassmannians by one dimensional torus

We consider the action of the one-parameter subgroup of the special linear group corresponding to a simple root on Grassmannians and describe the structure of the associated Geometric Invariant Theory (GIT) quotients with respect to Plücker line bundle. Using the combinatorics of Weyl group elements, we explicitly describe the semistable loci and identify cases where the resulting quotient admits the structure of a parabolic induction of a projective space. We further analyze the orbit structure under the Levi subgroup, compute the Picard group, connected component of the automorphism group and examine key geometric features such as Fano properties, cohomology of line bundles, and projective normality with respect to the descended linearization.

math.AG↗

arXiv2025

Character theory at a torsion element

The paper relates character value of an irreducible representation of a compact connected Lie group at certain elements of finite order with the dimension of a representation on another group, up to some precise constants, which all have significance. An important input is to analyse torsion elements of order d in an adjoint group with minimal dimensional centraliser, and to prove that in most cases when d divides the Coxeter number of G, this gives rise to a unique conjugacy class.

math.RT↗

arXiv2024

Mixed tensor invariants of Lie color algebra

In this paper, we consider the mixed tensor space of a $G$-graded vector space where $G$ is a finite abelian group. We obtain a spanning set of invariants of the associated symmetric algebra under the action of a color analogue of the general linear group which we refer to as the general linear color group. As a consequence, we obtain a generating set for the polynomial invariants, under the simultaneous action of the general linear color group, on color analogues of several copies of matrices. We show that in this special case, this is the set of trace monomials, which coincides with the set of generators obtained by Berele.

math.RT↗

arXiv2024

On tensor products of representations of Lie superalgebras

We consider typical finite dimensional complex irreducible representations of a basic classical simple Lie superalgebra, and give a sufficient condition on when unique factorization of finite tensor products of such representations hold. We also prove unique factorization of tensor products of singly atypical finite dimensional irreducible modules for $\mathfrak{sl}(m+1,n+1)$, $\mathfrak{osp}(2,2n)$, $G(3)$ and $F(4)$ under an additional assumption. This result is a Lie superalgebra analogue of Rajan's fundamental result \cite{MR2123935} on unique factorization of tensor products for finite dimensional complex simple Lie algebras.

math.RT↗

arXiv2023

Graded Picture Invariants and polynomial invariants for mixed tensor superspaces

In this paper we consider the mixed tensor space of a $\mathbb Z_2$-graded vector space. We obtain a spanning set of invariants of the associated symmetric algebra under the action of the general linear supergroup as well as the queer supergroup over the Grassmann algebra. As a consequence, we give a generating set of polynomial invariants for the simultaneous adjoint action of the general linear supergroup on several copies of its Lie superalgebra. We show that in this special case, these turn out to be the supertrace monomials which is analogous to the results of Procesi in the classical case. A queer supergroup analogue of these results is also obtained.

math.RT↗

arXiv2022

Quotients of commuting schemes associated to Symmetric Pairs

Let $\mathfrak{g}=\mathfrak{g}_0\oplus \mathfrak{g}_1$ be a $\mathbb Z_2$-grading of a classical Lie algebra such that $(\mathfrak{g}, \mathfrak{g}_0)$ is a classical symmetric pair. Let $G$ be a classical group with Lie algebra $\mathfrak{g}$ and let $G_0$ be the connected subgroup of $G$ with ${\rm Lie} (G_0)=\mathfrak g_0$. For $d \geq 2$, let $\mathfrak{C}^d(\mathfrak{g}_1)$ be the $d$-th commuting scheme associated with the symmetric pair $(\mathfrak g, \mathfrak g_0)$. In this article, we study the categorical quotient $\mathfrak{C}^d(\mathfrak{g}_1)//{G_0}$ via the Chevalley restriction map. As a consequence we show that the categorical quotient scheme $\mathfrak C^d(\mathfrak g_1)//G_0$ is normal and reduced. As a part of the proof, we describe a generating set for the algebra $k[\mathfrak{g}_1^d]^{G_0}$, which are of independent interest.

math.RT↗

arXiv2022

A note on branching of $V(ρ)$

Let $\mathfrak{g}$ be a complex simple Lie algebra and let $\mathfrak{g}_0$ be the sub-algebra fixed by a diagram automorphism of $\mathfrak{g}$. Let $G$ be the complex, simply-connected, simple algebraic group with Lie algebra $\mathfrak{g}$, and let $G_0$ be the connected subgroup of $G$ with Lie algebra $\mathfrak{g}_0$. Let $ρ$ be the half sum of positive roots of $\mathfrak{g}$. In this article, we give a necessary and sufficient condition for a highest weight $\mathfrak{g}_0$-representation $V_0(dμ)$ to occur in the representation ${\rm res}_{\mathfrak{g}_0}V(dρ)$, for any saturation factor $d$ of the pair $(G_0, G)$.

math.RT↗