Aleksandrs Mihailovs
The spaces of invariants of tensor powers of the defining representation of Sp(2n) are provided with the bases parametrized by symplectic wave graphs introduced here especially for this purpose. The proof utilizes a game similar to Tetris, named here S-tris. This work continues my previous work on the tensor invariants of SL(n), wave graphs and L-tris.
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The space of invariants of a tensor product of representations of SL(n) is provided with the basis parametrized by wave graphs introduced here especially for this purpose. The proof utilizes a game similar to Tetris, named here L-tris.
The symplectic blob algebras are a family of finite dimensional noncommutative algebras over $\mathbb{Z}[X_1,X_2,X_3,X_4,X_5,X_6]$ that can be defined in terms of planar diagrams in a way that extends the Temperley-Lieb and (ordinary) blob algebras. In this paper we construct a new "tensor space" representation of the symplectic blob algebra $\mathcal{A} \otimes \sba$ for each $n \in \mathbb{N}$, for $\mathcal{A}$ a particular commutative ring with indeterminates. The form of this representation is motivated by the XXZ representation of the Temperley-Lieb algebra $\TL_n$ \cite{jimbo86} and the…
We give two proofs of a level-rank duality for braided fusion categories obtained from quantum groups of type $C$ at roots of unity. The first proof uses conformal embeddings, while the second uses a classification of braided fusion categories associated with quantum groups of type $C$ at roots of unity. In addition we give a similar result for non-unitary braided fusion categories quantum groups of types $B$ and $C$ at odd roots of unity.
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 $Sp_{2n_1}\times\cdots\times Sp_{2n_k}$ where $n, n_1,\ldots, n_k$ are positive integers such that $n = n_1+\cdots+n_k$ and the $n_j$'s depend on $m$ as well as the representations in the tensor product. Using this result and a work of J. Lepowsky, we obtain a skew Pieri rule for $Sp_{2m}$, i.e., a description of the irreducible decomposition of the tensor product of an irreducible representa…
Let $\Field$ be an algebraically closed field. For $n \in \mathbb{N}$ and $δ, δ_L, δ_R, κ_L, κ_R, κ\in \Field$, the symplectic blob algebra $\sba(δ, δ_L, δ_R, κ_L, κ_R, κ)$ is a finite dimensional non-commutative $\Field$-algebra that may be viewed as an extension of the Temperley-Lieb algebra. In a previous paper, we defined, for any $n \in \mathbb{N}$, a tensor space module $\tensor[_\sba]{\mathcal{V}(n)}{}$. In this paper we generalise an argument used by Martin and Ryom-Hansen in their study of the (ordinary) blob algebra to show that when $\sba$ is quasihereditary the module $\tensor[_\sb…
We consider the fine grading of $sl(n,\mb C)$ induced by tensor product of generalized Pauli matrices in the paper. Based on the classification of maximal diagonalizable subgroups of $PGL(n,\mb C)$ by Havlicek, Patera and Pelantova, we prove that any finite maximal diagonalizable subgroup $K$ of $PGL(n,\mb C)$ is a symplectic abelian group and its Weyl group, which describes the symmetry of the fine grading induced by the action of $K$, is just the isometry group of the symplectic abelian group $K$. For a finite symplectic abelian group, it is also proved that its isometry group is always gene…
In this work we develop an integral representation for the partial $L$-function of a pair $π\timesτ$ of genuine irreducible cuspidal automorphic representations, $π$ of the $m$-fold covering of Matsumoto of the symplectic group $Sp_{2n}$, and $τ$ of a certain covering group of $GL_k$, with arbitrary $m$, $n$ and $k$. Our construction is based on the recent extension by Cai, Friedberg, Ginzburg and the author, of the classical doubling method of Piatetski-Shapiro and Rallis, from rank-$1$ twists to arbitrary rank twists. We prove a basic global identity for the integral and compute the local in…
Let $Σ_g$ be a closed oriented surface of genus g and let $H_\mathbb{Q}$ denote $H_1(Σ_g;\mathbb{Q})$ which we understand to be the standard symplectic vector space over $\mathbb{Q}$ of dimension $2g$. We introduce a canonical metric on the space $(H_\mathbb{Q}^{\otimes 2k})^{\mathrm{Sp}}$ of symplectic invariant tensors by analyzing the structure of the vector space $\mathbb{Q}\mathcal{D}^{\ell}(2k)$ generated by linear chord diagrams with $2k$ vertices. This space, equipped with a certain inner product, serves as a universal model for $(H^{\otimes 2k})^{\mathrm{Sp}}$ for any $g$. We decompos…
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 established. The relations are the so called P_n relations which appear in the study of certain 3-manifold invariants. Next a class of cohomology groups are introduced, called Pluri-Hodge groups (somewhat in keeping with the notion of pluri-canonical groups). These are Dolbeault groups on a complex manifold X with values in tensor powers of sheaves of holomorphic f…
The affine and degenerate affine Birman-Murakami-Wenzl (BMW) algebras arise naturally in the context of Schur-Weyl duality for orthogonal and symplectic quantum groups and Lie algebras, respectively. Cyclotomic BMW algebras, affine and cyclotomic Hecke algebras, and their degenerate versions are quotients. In this paper we explain how the affine and degenerate affine BMW algebras are tantalizers (tensor power centralizer algebras) by defining actions of the affine braid group and the degenerate affine braid algebra on tensor space and showing that, in important cases, these actions induce acti…
We consider the space of polydifferential operators on n functions on symplectic manifolds invariant under symplectic automorphisms, whose study was initiated by Mathieu in 1995. Permutations of inputs yield an action of S_n, which extends to an action of S_{n+1}. We study this structure viewing n as a parameter, in the sense of Deligne's category. For manifolds of dimension 2d, we show that the isotypic part of this space of <= 2d+1-th tensor powers of the reflection representation h=C^n of S_{n+1} is spanned by Poisson polynomials. We also prove a partial converse, and compute explicitly the…
In this paper we study the sp(2m)-invariant Dirac operator Ds which acts on symplectic spinors, from an orthogonal point of view. By this we mean that we will focus on the subalgebra so(m), as this will allow us to derive branching rules for the space of 1-homogeneous polynomial solutions for the operator Ds (hence generalising the classical Fischer decomposition in harmonic analysis for a vector variable in Rm). To arrive at this result we use techniques from representation theory, including the transvector algebra Z(sp(4),so(4)) and tensor products of Verma modules.
We give a classification of $1^{st}$ order invariant differential operators acting between sections of certain bundles associated to Cartan geometries of the so called metaplectic contact projective type. These bundles are associated via representations, which are derived from the so called higher symplectic, harmonic or generalized Kostant spinor modules. Higher symplectic spinor modules are arising from the Segal-Shale-Weil representation of the metaplectic group by tensoring it by finite dimensional modules. We show that for all pairs of the considered bundles, there is at most one $1^{st}$…
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