On the matrix units for the symmetric group
نویسنده
چکیده
We give a simple proof of the equivalence of the matrix unit formulas for the symmetric group provided by Murphy’s construction and by the fusion procedure due to Cherednik. 1 Young basis Let us fix some notation and recall some well known facts about the representations of the symmetric group Sn; see e.g. [6]. We write a partition λ as a sequence λ = (λ1, . . . , λl) of integers such that λ1 > · · · > λl > 0. We shall identify a partition λ with its diagram which is a left-justified array of rows of cells such that the top row contains λ1 cells, the next row contains λ2 cells, etc. Let us fix a positive integer n. If λ1 + · · ·+ λl = n then λ is a partition of n, written λ ⊢ n. A cell of λ is called removable if its removal leaves a diagram. Similarly, a cell is addable to λ if the union of λ and the cell is a diagram. We shall write μ → λ if λ is obtained from μ by adding one cell. A tableau T of shape λ (or a λ-tableau T ) is obtained by filling in the cells of the diagram with the numbers {1, . . . , n} so that each cell contains exactly one number. We write sh(T ) = λ if the shape of T is λ. A tableau T is called standard if its entries strictly increase along the rows and down the columns. The irreducible representations of Sn over C are parameterized by partitions of n. Given a partition λ of n denote the corresponding irreducible representation of Sn by Vλ. The vector space Vλ is equipped with an Sn-invariant inner product ( , ). The orthonormal Young basis {vT} of Vλ is parameterized by the set of standard λtableaux T . The action of the standard generators si = (i, i+ 1) of Sn in the Young
منابع مشابه
On the construction of symmetric nonnegative matrix with prescribed Ritz values
In this paper for a given prescribed Ritz values that satisfy in the some special conditions, we find a symmetric nonnegative matrix, such that the given set be its Ritz values.
متن کاملThe (R,S)-symmetric and (R,S)-skew symmetric solutions of the pair of matrix equations A1XB1 = C1 and A2XB2 = C2
Let $Rin textbf{C}^{mtimes m}$ and $Sin textbf{C}^{ntimes n}$ be nontrivial involution matrices; i.e., $R=R^{-1}neq pm~I$ and $S=S^{-1}neq pm~I$. An $mtimes n$ complex matrix $A$ is said to be an $(R, S)$-symmetric ($(R, S)$-skew symmetric) matrix if $RAS =A$ ($ RAS =-A$). The $(R, S)$-symmetric and $(R, S)$-skew symmetric matrices have a number of special properties and widely used in eng...
متن کاملSome results on the symmetric doubly stochastic inverse eigenvalue problem
The symmetric doubly stochastic inverse eigenvalue problem (hereafter SDIEP) is to determine the necessary and sufficient conditions for an $n$-tuple $sigma=(1,lambda_{2},lambda_{3},ldots,lambda_{n})in mathbb{R}^{n}$ with $|lambda_{i}|leq 1,~i=1,2,ldots,n$, to be the spectrum of an $ntimes n$ symmetric doubly stochastic matrix $A$. If there exists an $ntimes n$ symmetric doubly stochastic ...
متن کاملOn the numerical solution of generalized Sylvester matrix equations
The global FOM and GMRES algorithms are among the effective methods to solve Sylvester matrix equations. In this paper, we study these algorithms in the case that the coefficient matrices are real symmetric (real symmetric positive definite) and extract two CG-type algorithms for solving generalized Sylvester matrix equations. The proposed methods are iterative projection metho...
متن کاملIMPROVING THE SELECTION SYMMETRIC WEIGHTS AS A SECONDARY GOAL IN DEA CROSS-EFFICIENCY EVALUATION
Recently, some authors proposed the use of symmetric weightsfor computing the elements of cross-efficiency matrix. In spite ofthe fact that the proposed method decreases the number of zeroweights, a large number of zero weights may still exist among inputand output symmetric weights. To decrease the number of input andoutput symmetric weights, this paper improves the proposed secondarygoal mode...
متن کامل