A criterion on the semisimple Brauer algebras II

Abstract

In [H. Rui, A criterion on the semisimple Brauer algebras, J. Combin. Theory Ser. A 111 (2005) 78-88], the first author gave an algorithm for determining the pairs (n, δ) such that the Brauer algebra B n (δ) over a field F is semisimple. Such an algorithm involves a subset Z(n) ⊂ Z. In this note, we give an explicit description about Z(n). Using [H. Rui, A criterion on the semisimple Brauer algebras, J. Combin. Theory Ser. A 111 (2005) 78-88, 1.3] we verify Enyang's conjecture given in [J. Enyang, Specht modules and semisimplicity criteria for Brauer and Birman-Murakami-Wenzl algebras, preprint, 2005, 12.2].

The second fundamental theorem of invariant theory for the orthogonal group

“…The orthogonal group O(V ) is the isometry group of this form, defined as O(V ) = {g ∈ GL(V ) | (gv, gw) = (v, w) ∀ v, w ∈ V }. In [3], Brauer showed that the first fundamental theorem of invariant theory for O(V ) implies that there is a surjective map ν from the Brauer algebra B r (n) over K to End O(V ) (V ⊗r ), but the fact [4,19] that B r (n) is semisimple if and only if r ≤ n + 1, has complicated the determination of the kernel of ν, and therefore limited the use of this fact.…”

Morita equivalences on Brauer algebras and BMW algebras of simply-laced types

“…The Morita equivalences ( [21]) and Quasi-heredity ( [3]) are important properties of associative algebras. As cellular algebras( [14], [15], [16]), these properties of classical Brauer algebras and BMW algebras, even some related algebras, are well studied in many papers, such as König and Xi ([16], [17], [18], [29]), Rui and Si([23], [24], [25], [26], [27]). Their results are based on studying the bilinear forms for defining their cellular structures.…”

Gram determinants and semisimplicity criteria for Birman-Wenzl algebras

“…We can not restrict a B n -module to H n . In other words, we can not use the method in [16,17] to give a criterion for B n being semisimple. Finally, we remark that the method we use in the current paper can be used to deal with cyclotomic Nazarov-Wenzl algebras [1].…”

“…k=3 {q 3−2k , ±q 3−k , −q 2k−3 , ±q k−3 } and n ≥ 3. They are the parameters we got in [17] such that the corresponding Brauer algebra is not semisimple. Finally, we remark that some partial results on Brauer algebras being semisimple over C can be found in [5,6,20].…”