The Terwilliger algebra of symplectic dual polar graphs, the subspace lattices and \(U_q(\mathrm{sl}_2)\) (Q2675872)

In this paper, the authors present the construction of the symplectic dual polar graphs and describe some of their properties by looking at the automorphism groups of these graphs and their actions on the standard modules and highlighting the existence of a special abelian subgroup \(H\). They construct a basis of the standard module which diagonalizes the action of the group elements in \(H\) and show that there is a one-to-one correspondence between the vectors of this basis and the characteristic vectors of a subspace lattice \(L_{N}(q)\). The subspace lattice graph \(L_{N}(q)\) and its relation with the quantum group \(U_{\sqrt{q}}(\mathrm{sl}_2)\) are also presented. It is shown that the restriction of the adjacency matrix of a dual polar graph of type \([CD(q)]\) to an eigenspace of \(H\) corresponds to the adjacency matrix of a weighted subspace lattice. This connection and the relation between \(L_{N}(q)\) and \(U_{\sqrt{q}}(\mathrm{sl}_2)\) are used to obtain the irreducible \({\mathcal T}\)-submodules, where \({\mathcal T}\) is the Terwilliger algebra of symplectic dual polar graphs. The multiplicities of the isomorphic submodules are given.