Definition
The Brauer algebra depends on the choice of a positive integer n and a number d (which in practice is often the dimension of the fundamental representation of an orthogonal group Od). The Brauer algebra has dimension (2n)!/2nn! = (2n − 1)(2n − 3) ··· 5·3·1 and has a basis consisting of all pairings on a set of 2n elements X1, ..., Xn, Y1, ..., Yn (that is, all perfect matchings of a complete graph K2n: any two of the 2n elements may be matched to each other, regardless of their symbols). The elements Xi are usually written in a row, with the elements Yi beneath them. The product of two basis elements A and B is obtained by first identifying the endpoints in the bottom row of A and the top row of B (Figure AB in the diagram), then deleting the endpoints in the middle row and joining endpoints in the remaining two rows if they are joined, directly or by a path, in AB (Figure AB=nn in the diagram).
Read more about this topic: Brauer Algebra
Famous quotes containing the word definition:
“Scientific method is the way to truth, but it affords, even in
principle, no unique definition of truth. Any so-called pragmatic
definition of truth is doomed to failure equally.”
—Willard Van Orman Quine (b. 1908)
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)
“I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)