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Register a new userThe Terwilliger algebra of symplectic dual polar graphs, the subspace lattices and $U_q(sl_2)$
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The adjacency matrix of a symplectic dual polar graph restricted to the eigenspaces of an abelian automorphism subgroup is shown to act as the adjacency matrix of a weighted subspace lattice. The connection between the latter and $U_q(sl_2)$ is used to find the irreducible components of the standard module of the Terwilliger algebra of symplectic dual polar graphs. The multiplicities of the isomorphic submodules are given.
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Let $Dgeq 3$ denote an integer. For any $xin mathbb F_2^D$ let $w(x)$ denote the Hamming weight of $x$. Let $X$ denote the subspace of $mathbb F_2^D$ consisting of all $xin mathbb F_2^D$ with even $w(x)$. The $D$-dimensional halved cube $frac{1}{2}H(D,2)$ is a finite simple connected graph with vertex set $X$ and $x,yin X$ are adjacent if and only if $w(x-y)=2$. Fix a vertex $xin X$. The Terwilliger algebra $mathcal T=mathcal T(x)$ of $frac{1}{2}H(D,2)$ with respect to $x$ is the subalgebra of ${rm Mat}_X(mathbb C)$ generated by the adjacency matrix $A$ and the dual adjacency matrix $A^*=A^*(x)$ where $A^*$ is a diagonal matrix with $$ A^*_{yy}=D-2w(x-y) qquad hbox{for all $yin X$}. $$ In this paper we decompose the standard $mathcal T$-module into a direct sum of irreducible $mathcal T$-modules.
In this paper, we develop a new method for constructing $m$-ovoids in the symplectic polar space $W(2r-1,q)$ from some strongly regular Cayley graphs in cite{Brouwer1999Journal}. Using this method, we obtain many new $m$-ovoids which can not be derived by field reduction.
We consider orbit partitions of groups of automorphisms for the symplectic graph and apply Godsil-McKay switching. As a result, we find four families of strongly regular graphs with the same parameters as the symplectic graphs, including the one discovered by Abiad and Haemers. Also, we prove that switched graphs are non-isomorphic to each other by considering the number of common neighbors of three vertices.
In this article we associate a combinatorial differential graded algebra to a cubic planar graph G. This algebra is defined combinatorially by counting binary sequences, which we introduce, and several explicit computations are provided. In addition, in the appendix by K. Sackel the F(q)-rational points of its graded augmentation variety are shown to coincide with (q+1)-colorings of the dual graph.
We introduce and define the quantum affine $(m|n)$-superspace (or say quantum Manin superspace) $A_q^{m|n}$ and its dual object, the quantum Grassmann superalgebra $Omega_q(m|n)$. Correspondingly, a quantum Weyl algebra $mathcal W_q(2(m|n))$ of $(m|n)$-type is introduced as the quantum differential operators (QDO for short) algebra $textrm{Diff}_q(Omega_q)$ defined over $Omega_q(m|n)$, which is a smash product of the quantum differential Hopf algebra $mathfrak D_q(m|n)$ (isomorphic to the bosonization of the quantum Manin superspace) and the quantum Grassmann superalgebra $Omega_q(m|n)$. An interested point of this approach here is that even though $mathcal W_q(2(m|n))$ itself is in general no longer a Hopf algebra, so are some interesting sub-quotients existed inside. This point of view gives us one of main expected results, that is, the quantum (restricted) Grassmann superalgebra $Omega_q$ is made into the $mathcal U_q(mathfrak g)$-module (super)algebra structure,$Omega_q=Omega_q(m|n)$ for $q$ generic, or $Omega_q(m|n, bold 1)$ for $q$ root of unity, and $mathfrak g=mathfrak{gl}(m|n)$ or $mathfrak {sl}(m|n)$, the general or special linear Lie superalgebra. This QDO approach provides us with explicit realization models for some simple $mathcal U_q(mathfrak g)$-modules, together with the concrete information on their dimensions. Similar results hold for the quantum dual Grassmann superalgebra $Omega_q^!$ as $mathcal U_q(mathfrak g)$-module algebra.In the paper some examples of pointed Hopf algebras can arise from the QDOs, whose idea is an expansion of the spirit noted by Manin in cite{Ma}, & cite{Ma1}.
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