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 related Martin-Woodcock representation of the blob algebra $b_n$ \cite{martinwoodcock2003}. For $k$ an algebraically closed field, and for any $(δ,δ_L,δ_R,κ_L,κ_R,κ) \in k^{6}$, the algebra $\sba$ specialises to a $k$-algebra $\sba(δ,δ_L,δ_R,κ_L,κ_R,κ)$. For any such specialisation, our representation passes to a $\sba(δ,δ_L,δ_R,κ_L,κ_R,κ)$-module $\mathcal{V}(n)$.
This paper reports key advances in the study of the representation theory of the symplectic blob algebra. For suitable specialisations of the parameters we construct four large families of homomorphisms between cell modules. We hence find a large family of non-semisimple specialisations. We find a minimal poset (i.e. least number of relations) for the symplectic blob as a quasi-hereditary algebra.
The symplectic blob algebra $b_n$ ($n \in \mathbb{N}$) is a finite dimensional algebra defined by a multiplication rule on a basis of certain diagrams. The rank $r(n)$ of $b_n$ is not known in general, but $r(n)/n$ grows unboundedly with $n$. For each $b_n$ we define an algebra by presentation, such that the number of generators and relations grows linearly with $n$. We prove that these algebras are isomorphic.
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 $\tensor[_\sb…
This paper reports some advances in the study of the symplectic blob algebra. We find a presentation for this algebra. We find a minimal poset for this as a quasi-hereditary algebra. We discuss how to reduce the number of parameters defining the algebra from 6 to 4 (or even 3) without loss of representation theoretic generality. We then find some non-semisimple specialisations by calculating Gram determinants for certain cell modules (or standard modules) using the good parametrisation defined. We finish by considering some quotients of specialisations of the symplectic blob algebra which are …
The recollement approach to the representation theory of sequences of algebras is extended to pass basis information directly through the globalisation functor. The method is hence adapted to treat sequences that are not necessarily towers by inclusion, such as symplectic blob algebras (diagram algebra quotients of the type-$\hati{C}$ Hecke algebras). By carefully reviewing the diagram algebra construction, we find a new set of functors interrelating module categories of ordinary blob algebras (diagram algebra quotients of the type-${B}$ Hecke algebras) at {\em different} values of the algebra…
We show that the Ariki-Terasoma-Yamada tensor module and its permutation submodules $ M(λ) $ are modules for the blob algebra when the Ariki-Koike algebra is a Hecke algebra of type $B$. We show that $ M(λ)$ and the standard modules $ Δ(λ) $ have the same dimensions, the same localization and similar restriction properties and are equal in the Grothendieck group. Still we find that the universal property for $ Δ(λ) $ fails for $ M(λ) $, making $ M(λ) $ and $ Δ(λ) $ different modules in general. Finally, we prove that $ M(λ) $ is isomorphic to the dual Specht module for the Ariki-Koike algebra.
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…
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…
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.
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.
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…
We study some algebraic and combinatorial features of two algebras that arise as quotients of Temperley-Lieb algebras of type $\tilde{C}$, namely, the two-boundary Temperley-Lieb algebra and the symplectic blob algebra. We provide a monomial basis for both algebras. The elements of these bases are parameterized by certain subsets of fully commutative elements. We enumerate these elements according to their affine length.
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…
The character formula of any finite dimensional irreducible module $L_λ$ for Lie superalgebra $\mathfrak{osp}(n|2)$ is computed. As a by-product, the decomposition of tensor module $L_λ\otimes \mathbb{C}^{n|2}$, where $\mathbb{C}^{n|2}$ is the natural representation, is obtained.
We give the first two-dimensional pictorial presentation of Berele's correspondence \cite{Berele}, an analogue of the Robinson-Schensted (R-S) correspondence \cite{Robinson, Schensted} for the symplectic group $Sp(2n, \Cpx )$. From the standpoint of representation theory, the R-S correspondence combinatorially describes the irreducible decomposition of the tensor powers of the natural representation of $GL(n,\Cpx)$. Berele's insertion algorithm gives the bijection that describes the irreducible decomposition of the tensor powers of the natural representation of $Sp(2n, \Cpx)$. Two-dimensional …
A Chevalley type integral basis for the ortho-symplectic Lie superalgebra is constructed. The simple modules of the ortho-symplectic supergroup over an algebraically closed field of prime characteristic not equal to 2 are classified, where a key combinatorial ingredient comes from the Mullineux conjecture on modular representations of the symmetric group. A Steinberg tensor product theorem for the ortho-symplectic supergroup is also obtained.
The descent set of an oscillating (or up-down) tableau is introduced. This descent set plays the same role in the representation theory of the symplectic groups as the descent set of a standard tableau plays in the representation theory of the general linear groups. In particular, we show that the descent set is preserved by Sundaram's correspondence. This gives a direct combinatorial interpretation of the branching rules for the defining representations of the symplectic groups; equivalently, for the Frobenius character of the action of a symmetric group on an isotypic subspace in a tensor po…
In this paper we prove Schur-Weyl duality between the symplectic group and Brauer algebra over an arbitrary infinite field $K$. We show that the natural homomorphism from the Brauer algebra $B_n(-2m)$ to the endomorphism algebra of tensor space $(K^{2m})^{\otimes n}$ as a module over the symplectic similitude group $GSp_{2m}(K)$ (or equivalently, as a module over the symplectic group $Sp_{2m}(K)$) is always surjective. Another surjectivity, that of the natural homomorphism from the group algebra for $GSp_{2m}(K)$ to the endomorphism algebra of $(K^{2m})^{\otimes n}$ as a module over $B_n(-2m)$…
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…
We show that the centraliser of the maximal compact subgroup of the real orthogonal or symplectic groups acting on tensors of their standard representation are isomorphic to cyclotomic Brauer algebras. We also show that for the symplectic group this cyclotomic Brauer algebra splits into summands of walled Brauer algebras.
We prove a Schur-Weyl duality between the quantum enveloping algebra of $\mathfrak{gl}_m$ and certain quotient algebras of Ariki-Koike algebras, which we give explicitly. The duality involves several algebraically independent parameters and is realized through the tensor product of a parabolic universal Verma module and a tensor power of the natural representation of $\mathfrak{gl}_m$. We also give a new presentation by generators and relations of the generalized blob algebras of Martin and Woodcock as well as an interpretation in terms of Schur-Weyl duality by showing they occur as a particul…
We construct in projective differential geometry of the real dimension $2$ higher symmetry algebra of the symplectic Dirac operator ${D}\kern-0.5em\raise0.22ex\hbox{/}_s$ acting on symplectic spinors. The higher symmetry differential operators correspond to the solution space of a class of projectively invariant overdetermined operators of arbitrarily high order acting on symmetric tensors. The higher symmetry algebra structure corresponds to a completely prime primitive ideal having as its associated variety the minimal nilpotent orbit of $\mathfrak{sl}(3,{\mathbb{R}})$.
Results on symplectic spinors and their higher spin versions, concerning representation theory and cohomology properties are presented. Exterior forms with values in the symplectic spinors are decomposed into irreducible modules including finding the hidden symmetry (Schur--Weyl--Howe type duality) given by a representation of the Lie superalgebra $\mathfrak{osp}(1|2)$ in this case. We also determine ranges of the induced exterior symplectic spinor derivatives when they are restricted to bundles induced by the irreducible submodules mentioned. This duality is used to decompose curvature tensor…
We introduce the spinor representations for osp(m|2n). These generalize the spinors for so(m) and the symplectic spinors for sp(2n) and correspond to representations of the supergroup with supergroup pair (Spin(m) x Mp(2n),osp(m|2n)). We prove that these spinor spaces are uniquely characterized as the completely pointed osp(m|2n)-modules. Then the tensor product of this representation with irreducible finite dimensional osp(m|2n)-modules is studied. Therefore we derive a criterion for complete reducibility of tensor product representations. We calculate the decomposition into irreducible osp(m…
