Related papers: Symplectic determinant laws and invariant theory

We use the theory of skew duality to show that decomposing the tensor product of $k$ irreducible representations of the symplectic group $Sp_{2m} = Sp_{2m}(C)$ is equivalent to branching from $Sp_{2n}$ to $Sp_{2n_1}\times\cdots\times…

Let $(Q,\sigma)$ be a symmetric quiver, where $Q=(Q_0,Q_1)$ is a finite quiver without oriented cycles and $\sigma$ is a contravariant involution on $Q_0\sqcup Q_1$. The involution allows us to define a nondegenerate bilinear form $<,>$ on…

If $A$ is a $2n \times 2n$ real positive definite matrix, then there exists a symplectic matrix $M$ such that $M^TAM = \left [ \begin{array}{cc} D & O \\ O & D \end{array} \right ]$ where $D= \diag (d_1 (A), \ldots, d_n(A))$ is a diagonal…

The variety of principal minors of $n\times n$ symmetric matrices, denoted $Z_{n}$, is invariant under the action of a group $G\subset \GL(2^{n})$ isomorphic to $\G$. We describe an irreducible $G$-module of degree $4$ polynomials…

A basis for each finite-dimensional irreducible representation of the symplectic Lie algebra sp(2n) is constructed. The basis vectors are expressed in terms of the Mickelsson lowering operators. Explicit formulas for the matrix elements of…

This article deals with a number of topics which are, somewhat surprisingly, related. Firstly, the fundamental theorem of skew invariant theory for the symplectic group giving the generators and relations of symplectic invariants is…

We provide a short proof of the theorem that every real multivariate polynomial has a symmetric determinantal representation, which was first proved in J. W. Helton, S. A. McCullough, and V. Vinnikov, Noncommutative convexity arises from…

We provide an explicit algorithm to calculate invariant tensors for the adjoint representation of the simple Lie algebra $sl(n)$, as well as arbitrary representation in terms of roots. We also obtain explicit formulae for the adjoint…

We define a GL-variety to be a (typically infinite dimensional) algebraic variety equipped with an action of the infinite general linear group under which the coordinate ring forms a polynomial representation. Such varieties have been used…

For every $2n\times 2n$ real positive definite matrix $A,$ there exists a real symplectic matrix $M$ such that $M^TAM=\diag(D,D),$ where $D$ is the $n\times n$ positive diagonal matrix with diagonal entries $d_1(A)\le \cdots\le d_n(A).$ The…

A fundamental problem from invariant theory is to describe the endomorphism algebra of multilinear functions on a representation V invariant under the action of a group G. According to Weyl's classic, a first main (later: fundamental)…

Much has been written on reciprocity laws in number theory and their connections with group representations. In this paper we explore more on these connections. We prove a "reciprocity Law" for certain specific representations of semidirect…

We consider a generalization of representations of quivers that can be derived from the ordinary representations of quivers by considering a product of arbitrary classical groups instead of a product of the general linear groups and by…

For any Lie group $G$, we construct a $G$-equivariant analogue of symplectic capacities and give examples when $G = \mathbb{T}^k\times\mathbb{R}^{d-k}$, in which case the capacity is an invariant of integrable systems. Then we study the…

We define higher genus Gromov-Witten invariants and establish a mathematical theory of sigma model coupled with gravity over any semi-positive symplectic manifolds. As applications, we verify the stablizing conjecture of symplectic…

We prove the Chevalley restriction theorem for the commuting scheme of symplectic Lie algebras. The key step is the construction of the inverse map of the Chevalley restriction map called the spectral data map. Along the way, we establish a…

Let G be a finite group acting on a symplectic complex vector space V. Assume that the quotient V/G has a holomorphic symplectic resolution. We prove that G is generated by "symplectic reflectionsd"', i.e. symplectomorphisms with fixed…

We introduce the notion of a subregular subalgebra, which we believe is useful for classification of subalgebras of Lie algebras. We use it to construct a non-regular invariant generalized complex structure on a Lie group. As an…

In this paper we consider the restriction of a unitary irreducible representation of type $A_{\mathfrak q}(\lambda)$ of $GL(4,{\mathbb R})$ to reductive subgroups $H$ which are the fixpoint sets of an involution. We obtain a formula for the…

We introduce the notion of the symplectic characteristic polynomial of an endomorphism of a symplectic vector space. This is a polynomial in two variables and can be considered as a generalization of the characteristic polynomial of the…