Brauer Algebras and Centralizer Algebras for SO(2n,C)

Abstract

In 1937, Richard Brauer identified the centralizer algebra of transformations Ž . commuting with the action of the complex special orthogonal groups SO 2 n .Ž . Ž mk . nCorresponding to the centralizer algebra E 2 n s End

Integration with Respect to the Haar Measure on Unitary, Orthogonal and Symplectic Group

“…Its invariant theory has first been studied by R. Brauer [Bra37] who introduced a family of algebras, nowadays called Brauer algebras. These algebras have been at the center of many investigations (see [BW89,Gro99] and the references therein). Some actions of these algebras lead to an analogue of the Schur-Weyl duality in the case of the orthogonal group and symplectic groups and for this reason they are very useful for our purposes.…”

Schur–Weyl duality for the Brauer algebra and the ortho-symplectic Lie superalgebra

“…This does not contradict Theorem 5.5 since in the case of m, n = 0 we have m(2n + 1) = 2mn + m which is strictly greater than the assumed bound in the theorem and in the case of n = 0 it holds m(2n + 1) = m which is also strictly greater. In the classical cases, the bounds are accurate for the symplectic and even orthogonal cases, but in the odd orthogonal case the Schur-Weyl duality holds already for 2m + 1 ≥ d, see [5, p. 870 Theorem b)] and [14,Theorem 1.4], whereas our bound is m ≥ d. Remark 5.9 Similar to the definition of the oriented Brauer category OB(m − n), one defines the Brauer category Br (δ) using Brauer diagrams having different numbers of vertices at top and bottom. Both, the embedding into the additive closure of the oriented Brauer category as well as the proof of surjectivity generalise to this situation, as they only rely on the diagrammatic description.…”

Webs and $q$-Howe dualities in types $\mathbf {BCD}$

“…(Morally, one "Brauer diagram generator" is missing for SO n if n is even, see also [Gro99] and [LZ16].) As a consequence, surjectivity fails in general for SO n in type D.…”