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nη²IΎ (kεwjuThe Terwilliger algebras of the Grassmann graphsv

The Terwilliger algebras of Q-polynomial distance-regular graphs are introduced by P.Terwilliger (1992,1993) and generalized by H.Suzuki (2005).

In this talk, we focus on the (generalized) Terwilliger algebra of the Grassmann graph and compute all the irreducible modules of it by considering the lattice X of subspaces of a finite-dimensional vector space over a finite field rather than the vertex set of the Grassmann graph, which consists of the subspaces of a fixed dimension.

The action on X of the group of linear transformations stabilizing a fixed subspace, brings about a two-parameter partition X_{i,j} of X. Then we define a complex matrix algebra with rows/columns indexed by X generated by the "raising" matrices R_1, R_2 and the "lowering" matrices L_1, L_2 with respect to the partition X_{i,j}. We show that the elements of the Terwilliger algebra are principal submatrices of those of this matrix algebra. We also compute all the irreducible modules of this matrix algebra by calculating the relations of the generating matrices.

Finally, restricting the lattice X to the vertex set of the Grassmann graph gives all the irreducible modules of the Terwilliger algebra.