arXiv subjects

Ian M. Musson

Publications and source records attributed to Ian M. Musson.

At least 19 recordsLinked to original sources

arXiv2026

Geometry and coefficients of Šapovalov elements for KM Lie superalgebras

We study Shapovalov elements for a symmetrizable, integrable Kac-Moody Lie superalgebra $\mathfrak{g}$. Let $γ$ be a positive root of $\mathfrak{g}$ and $m$ a positive integer, with suitable conditions on the pair $(γ, m).$ The Shapovalov element $θ_{γ,m}\in U(\mathfrak{b}^-) $ has the important property that if $λ$ lies on a certain hyperplane, then $θ_{γ, m} v_λ$ is a highest weight vector of weight $λ-mγ$ in $M(λ)$. We give a closed fomula for all coefficients that arise in the induction step of the proof. The commutative version of this formula relates the leading terms using maps graded algebras that we call of Shapovalov algebras. This suggests a geometric setting for the study of Shapovalov elements.

math.RT↗

arXiv2026

Catalan numbers and a conjecture on the maximum composition length of a Kac module

Let $f:\mathbb{Z}\longrightarrow \{ \times \cdot\}$ be a function such that $f(a) = \cdot$ for all except finitely for many $a \in \mathbb{Z}$. We define a set $\flat f$ of non-intersecting arc (or cap) diagrams satisfying certain conditions determined by $f$. Then we give a recursive method for enumeration of $\flat f$ which recalls the Fundamental Recurrence for Catalan numbers. The motivation comes from the problem of enumeration of the composition factors of a Kac module with maximum degree of atypicality for the Lie superalgebra $\mathfrak{g}=\mathfrak{gl}(r|r)$. In particular we prove a conjecture that the maximum number of composition factors is a Catalan number.

math.CO↗

arXiv2025

Categorical quotients for actions of groupoids on varieties

For certain actions of the Weyl groupoid $\mathfrak{W}$ from [Sergeev and Veselov, Grothendieck rings of basic classical Lie superalgebras, Ann Math, 2011] on an affine variety $X$, geometric properties of the map $π: X \longrightarrow Y= {\operatorname{Spec }\;} \mathcal{O}(X)^\mathfrak{W}$ were studied in [Musson, On the geometry of some algebras related to the Weyl groupoid, Contemp. Math. 2024], In this paper we show that if the base field ${\mathtt k}$ is uncountable, the map $π$ is a geometric quotient which is universal in the category of ${\mathtt k}$-schemes. To do this we adapt a result from [{Mumford}, {Fogarty}, {Kirwan}, {1994}], showing that a geometric quotient is universal in the category of ${\mathtt k}$-schemes, to quotients by groupoids and more generally by equivalence relations. In our approach a key role is played by the closed points and Jacobson schemes.

math.AG↗

arXiv2024

Young diagrams, deformed Calogero-Moser systems and Cayley graphs

Let ${\mathtt{k}}$ be an algebraically closed field of characteristic zero and $n, m$ coprime positive integers. Let ${\stackrel{\rm o}{\mathfrak{g}}}$ be the Lie superalgebra ${\mathfrak{gl}}(n|m)$ with root system $Δ$. Using $Δ$, Sergeev and Veselov, \cite{SV2} introduced an action of the Weyl groupoid ${\mathcal{W}}$, in connection with their study of the the Grothendieck group of finite dimensinonal graded $\mathfrak{g}$-modules. We denote the subgroupoid of ${\mathcal{W}}$ with morphisms corresponding to isotropic roots by $\mathfrak T_{iso}$. Later, \cite{SV101} the same authors defined an action of ${\mathcal{W}}$ on $X={\mathtt{k}}^{n|m}$ such that the invariant algebra ${\mathcal{O}}(X)^{\mathcal{W}}$ is isomorphic to the algebra of quantum integrals for the deformed Calogero-Moser system introduced in \cite{SV1}. This completely integrable system depends on a non-zero parameter $κ$. When $κ=-m/n$ we study a certain infinite $\mathfrak T_{iso}$-orbit {\bf O} for this action. %which appears in \cite{SV101} Equation (14). The Cayley graph for this orbit is isomorphic to the Cayley graphs for two other actions of $\mathfrak T_{iso}$ which were studied in \cite{M23}.

math.RT↗

arXiv2024

Young diagrams, Borel subalgebras and Cayley graphs

Let $\mathtt{k}$ be an algebraically closed field of characteristic zero and $n, m$ coprime positive integers. Let ${\stackrel{\rm o}{\mathfrak{g}}}$ be the Lie superalgebra ${\mathfrak{sl}}(n|m)$ and let $\mathfrak T_{iso}$ be the groupoid introduced by Sergeev and Veselov \cite{SV2} with base the set of odd roots of ${\stackrel{\rm o}{\mathfrak{g}}}$. We show the Cayley graphs for three actions of $\mathfrak T_{iso}$ are isomorphic, These actions originate in quite different ways. Consider the set $X$ of Young diagrams contained in a rectangle with $n$ rows and $m$ columns. By adding or deleting rows and columns from certain diagrams and keeping track of the total number of boxes added or deleted, we obtain an equivalence relation on $X\times {\mathbb Z}$ such that $\mathfrak T_{iso}$ acts on the set of equivalence classes $[X\times {\mathbb Z}]$. We compare the action on $[X\times {\mathbb Z}]$ to an action on Borel subalgebras of the affinization ${\widehat{L}(\stackrel{\rm _o}{\mathfrak{g}})}$ of ${\stackrel{\rm o}{\mathfrak{g}}}$ which are related by odd reflections. The third action comes from an action of $\mathfrak T_{iso}$ on $\mathtt{k}^{n|m}$ defined by Sergeev and Veselov, motivated by deformed quantum Calogero-Moser problems \cite{SV1}. This action will be considered in \cite{M24}.

math.RT↗

arXiv2023

The Weyl groupoid in Type A, Young diagrams and Borel subalgebras

Let $\mathtt{k}$ be an algebraically closed field of characteristic zero. Let ${\stackrel{\rm o}{\mathfrak{g}}}$ be the Lie superalgebra ${\mathfrak{sl}}(n|m)$ and let $\mathfrak{W}$ be the Weyl groupoid introduced by Sergeev and Veselov using the root system of ${\stackrel{\rm o}{\mathfrak{g}}}$. An important subgroupoid $\mathfrak T_{iso}$ of ${\mathfrak{W}}$ has base $Δ_{iso}$, the set of all the isotropic roots. Motivated by deformed quantum Calogero-Moser problems, the same authors considered an action of $\mathfrak{W}$ on $\mathtt{k}^{n|m}$ depending on a parameter $κ$. %When $κ$ is negative special, they showed this action has infinite orbits. In the case $m>n$, with $m,n $ relatively prime and $κ=-n/m$ we study a particular infinite orbit of $\mathfrak T_{iso}$ with some special properties. This orbit, thought of as a directed graph is isomorphic to the graph of an orbit for the action of $\mathfrak T_{iso}$ on certain Borel subalgebras of the affinization ${\widehat{L}(\stackrel{\rm _o}{\mathfrak{g}})}$ of ${\stackrel{\rm o}{\mathfrak{g}}}$. %The root groupoid has a base consisting of Borel subalgebras with fixed even part, and morphisms are given by odd reflections. The underlying reason for this graph isomorphism is that both have combinatorics which can be described using Young diagrams and tableaux drawn on the surface of a rotating cylinder with circumference $n$ and length $m$. Allowing the cylinder to rotate produces an infinite orbit. This leads to a third graph which is isomorphic to the other two.

math.CO↗

arXiv2023

On the geometry of algebras related to the Weyl groupoid

Let $\mathtt{k}$ be an algebraically closed field of characteristic zero. Let $\mathfrak{g} $ be a finite dimensional classical simple Lie superalgebra over $\mathtt{k}$ or $\mathfrak{g} l(m,n)$. In the case that $\mathfrak{g} $ is a Kac-Moody algebra of finite type with set of roots $Δ$, Sergeev and Veselov introduced the Weyl groupoid $\mathfrak{W}=\mathfrak{W}(Δ)$, which has significant connections with the representation theory of $\mathfrak{g} $. Let $\mathfrak{h}$, $W$ and $Z(\mathfrak{g} )$ be a Cartan subalgebra of $\mathfrak{g} _0$, the Weyl group of $\mathfrak{g} _0$ and the center of $U(\mathfrak{g} )$ respectively. Also let $G$ be a Lie supergroup with Lie $G =\mathfrak{g} $. There are several important commutative algebras related to $\mathfrak{W}$. Namely \begin{itemize} \item The image $I(\mathfrak{h} )$ of the injective Harish-Chandra map $Z(\mathfrak{g} ){\longrightarrow} S(\mathfrak{h} )^W$. \item The supercharacter $\mathbb Z$-algebras $J(\mathfrak{g} )$ and $J(G)$ of finite dimensional representations of $\mathfrak{g} $ and $G$. \end{itemize} Let $\mathcal A = \mathcal A(\mathfrak{g})$ be denote either $I(\mathfrak{h} )$ or $J(G) \otimes_{\mathbb Z}{\mathtt{k}}$. The purpose of this paper is to investigate the algebraic geometry of $\mathcal A.$ In many cases, the algebra $\mathcal A$ satisfies the Nullstellensatz. This gives a bijection between radical ideals in $\mathcal A$ and superalgebraic sets (zero loci of such ideals). Any superalgebraic set is uniquely a finite union of irreducible superalgebraic components. In the non-exceptional Kac-Moody case, we describe the smallest superalgebraic set containing a given (Zariski) closed set, and show that the superalgebraic sets are exactly the closed sets that are unions of groupoid orbits.

math.RT↗

arXiv2022

Explicit expressions for \vSapovalov elements in Type A

We give explicit expressions for \vSapovalov elements in Type A Lie algebras and superalgebras. Explicit expressions were already given in arXiv:1710.10528 Section 9, using non-commutative determinants, and in fact our first main results, Theorems 2.3 and 2.6 can be viewed as complete expansions of these determinants. But we give new proofs, which seem easier because they avoid induction and cofactor expansion. We also describe \vSapovalov elements for fgl(m,n)with respect to an arbitrary Borel subalgebra in Theorem 3.7, and interpret \vSapovalov elements in Type A as determinants of Hessenberg matrices in Theorems 4.5 and 4.9.The exact form of the explicit expressions depends on an ordering on the set of positive roots, and Hessenberg matrices are useful in changing the ordering. Having explicit expressions for \vSapovalov elements allows us to give easy proofs of several results in representation theory, see Section 6 and Subsection 4.6.

math.RT↗

arXiv2022

Weyl groupoids and superalgebraic sets

This paper is a contribution to the study of the geometry of algebras related the Weyl groupoid initiated in \cite{M22}. The Nullstellensatz gives a bijection between radical ideals of such an algebra and their zero loci, the superalgebraic sets. Such sets are exactly the (Zariski) closed sets that are invariant under the action of a suitable groupoid, and the smallest superalgebraic set containing a given closed set can be described explicitly. Here we give several examples of superalgebraic sets. We also give several characterizations of Laurent supersymmetric polynomials. These adapt to unite several definitions of one of the algebras of interest, $J(G)$ that may be found in the literature.

math.RT↗

arXiv2022

Shapovalov Elements For $U_q(\mathfrak{sl}(N+1))

For a simple Lie algebra, Shapovalov elements give rise to highest weight vectors in Verma modules. The usual construction of these elements uses induction on the length of a certain Weyl group element. If $\mathfrak{g}= \mathfrak{sl}(N+1)$ explicit expressions for Shapovalov elements were given in [Mus22a]. Here we adapt the argument to the quantized enveloping algebra of $\mathfrak{g}$.

math.RT↗

arXiv2022

How to Construct the Lattice of Submodules of a Multiplicity free Module from Partial Information

In general it is a difficult problem to construct the lattice of submodules $L(M)$ of a given module $M$. In \cite{St} R. P. Stanley outlined a method for constucting a distributive lattice from a knowledge of its join irreducibles. However it is not an easy task to identify all join irreducible submodules of a given module. In the case of a multiplicity free module $M$ we present a modifiiction of Stanley's method based on the composition factors of $M$. As input we require a set of submodules $A_1,\ldots , A_n$ whose submodule lattices are known and which contain all composition factors of $M$. From this we can reconstruct $L(M)$. We illustrate the process for a family of Verma modules $M(\gl_n)$, with $n $ a positive integer, for the Lie superalgebra $\osp(3,2)$. We show that for $n\ge 2$, $L(M(\gl_n))$ is isomorphic to the (extended) free distributive lattice of rank 3.

math.RT↗

arXiv2019

The Nullstellensatz for supersymmetric polynomials

In this paper we prove a Nullstellensatz for supersymmetric polynomials. This gives a bijection between radical ideals and superalgebraic sets. These are algebraic sets which are invariant under the Weyl groupoid of Sergeev and Veselov, \cite{SV2}. Note that the algebra of supersymmetric polynomials is not Noetherian, so the usual Nullstellensatz does not apply. However it deos satisfy the ascending chain condition on radical ideals and this allows for the decomposition of superalgebraic sets into irreducible components. Analogous results hold for the a ring of Laurent supersymmetric polynomials. As an application, we give a proof of conjecture 13.5.1 from \cite{M}. This concerns the maximal ideals in the enveloping algebra of the general linear and orthosymplectic Lie superalgebras. The center is closely related to the algebra of supersymmetric polynomials and the result can be thought of as an analog of the weak Nullstellensatz.

math.RA↗

arXiv2018

Twisting Functors and Generalized Verma modules

Let $\mathfrak{g}$ be a reductive Lie algebra. We give a condition that ensures that the character of a generalized Verma module is well-behaved under a twisting functor. We show that a similar result holds for basic classical simple Lie superalgebras, and generalize a result from \cite{CM} about twisting Verma modules.

math.RT↗

arXiv2017

\vS{a}povalov elements and the Jantzen sum formula for contragredient Lie superalgebras

If $\mathfrak{g}$ is a contragredient Lie superalgebra and $γ$ is a root of $\mathfrak{g},$ we prove the existence and uniqueness of Šapovalov elements for $γ$ and give upper bounds on the degrees of their coefficients. Then we use Šapovalov elements to define some new highest weight modules. If $X$ is a set of orthogonal isotropic roots and $λ\in \mathfrak{h}^*$ is such that $λ+ρ$ is orthogonal to all roots in $X$, we construct a highest weight module $M^X(λ)$ with character $ε^λ{p}_X$. Here $p_X$ is a function that counts partitions not involving roots in $X$. Examples of such modules can be constructed via parabolic induction provided $X$ is contained in the set of simple roots of some Borel subalgebra. However our construction works without this condition and provides a highest weight module for the distinguished Borel subalgebra. The main results are analogs of the Šapovalov determinant and the Jantzen sum formula for $M^X(λ)$ when $\mathfrak{g}$ has type A.For the proof it is enough to study the behavior for certain "relatively general" highest weights. Using an equivalence of categories due to Cheng, Mazorchuk and Wang, the information we require is deduced from the behavior of the modules $M^X(λ)$ when $\mathfrak{g}=\mathfrak{gl}(2,1)$ or $\mathfrak{gl}(2,2)$. These low dimensional cases are studied in detail in an appendix.

math.RT↗

arXiv2015

The primitive spectrum for gl(m|n)

We study inclusions between primitive ideals in the universal enveloping algebra of general linear superalgebras. For classical Lie superalgebras, any primitive ideal is the annihilator of a simple highest weight module. It therefore suffices to study the quasi-order on highest weights determined by the relation of inclusion between primitive ideals. For the specific case of reductive Lie algebras, this quasi-order is essentially the left Kazhdan-Lusztig quasi-order. For Lie superalgebras, the classification is unknown in general, safe from some low dimensional specific cases. We derive an alternative definition of the left Kazhdan-Lusztig quasi-order which extends to classical Lie superalgebras. We show that a relation in this preorder implies an inclusion between primitive ideals. For gl(m|n) the new quasi-order is defined explicitly in terms of Brundan's Kazhdan-Lusztig theory. We prove that the quasi-order induces an actual partial order on the set of primitive ideals. We conjecture that this is the inclusion order. By the above paragraph one direction of this conjecture is true. We prove several consistency results concerning the conjecture and prove it for singly atypical and typical blocks of gl(m|n) and in general for gl(2|2). An important tool is a new translation principle for primitive ideals, based on the crystal structure for category O. Finally we focus on an interesting explicit example; the poset of primitive ideals contained in the augmentation ideal for gl(m|1).

math.RT↗

arXiv2015

Coefficients of Šapovalov elements for simple Lie algebras and contragredient Lie superalgebras

We provide upper bounds on the degrees of the coefficients of Šapovalov elements for a simple Lie algebra. If $\fg$ is a contragredient Lie superalgebra and $\gc$ is a positive isotropic root of $\fg,$ we prove the existence and uniqueness of the Šapovalov element for $\gc$ and we obtain upper bounds on the degrees of their coefficients. For type A Lie superalgebras we give a closed formula for Šapovalov elements. Often the coefficients of Šapovalov elements are products of linear factors, and we provide some reasons for this coming from representation theory. We also explore the relationships between Šapovalov elements coming from different roots, and their behavior when the Borel subalgebra is changed.

math.RT↗

arXiv2015

Sapovalov elements and the Jantzen filtration for contragredient Lie superalgebras: A Survey

This is a survey of some recent results on Sapovalov elements and the Jantzen filtration for contragredient Lie superalgebras. The topics covered include the existence and uniqueness of the Sapovalov elements, bounds on the degrees of their coefficients and the behavior of Sapovalov elements when the Borel subalgebra is changed. There is always a unique term whose coefficient has larger degree than any other term. This allows us to define some new highest weight modules. If X is a set of orthogonal isotropic roots and $λ\in h^*$ is such that $λ+ρ$ is orthogonal to all roots in X, we construct highest weight modules with character $ε^λp_X$. Here $p_X$ is a partition function that counts partitions not involving roots in X. When |X|=1, these modules are used to give a Jantzen sum formula for Verma modules in which all terms are characters of modules in the category O with positive coefficients.

math.RT↗

arXiv2014

Šapovalov elements for simple Lie algebras and basic classical simple Lie superalgebras

Let $M(\gl)$ be a Verma module for a basic classical simple Lie superalgebra $\fg \neq G(3)$ defined using the distinguished Borel subalgebra, and let $\gc$ be an isotropic positive root of $\fg.$ As a special case of our first main result we show that if $μ, \gl \in \fh^*$ with $\gl-μ= \gc$ we have $$\dim \Hom_{\sfg}(M(μ),M(\gl))\le 1.$$ This result applies to the construction of Šapovalov elements for isotropic roots. The proof rests on a comparison with the corresponding result for a certain simple Lie algebra $G$.

math.QA↗