Split bases and multiplicity separations in symmetric group transformation coefficients

Transformation coefficients between standard bases for irreducible representations of the symmetric group Sn and split bases adapted to the Sn 1 × Sn 2 ⊂ Sn subgroup (n 1 + n 2 = n) are considered. We first provide a selection rule and an identity rule for the subduction coefficients which allow to decrease the number of unknowns and equations arising from the linear method by Pan and Chen. Then, using the reduced subduction graph approach, we may look at higher multiplicity instances. As a significant example, an orthonormalized solution for the first multiplicity-three case, which occurs in the decomposition of the irreducible representation [4, 3, 2, 1] of S 10 into [3, 2, 1] ⊗ [3, 1] of S 6 × S 4 , is presented and discussed.

Section: Discussionmentioning

confidence: 99%

Section: Subduction Coefficients and Graphsmentioning

confidence: 99%

A reduced subduction graph and higher multiplicity inS_ntransformation coefficients

Chilla¹

2006

“…where (Q k ) ρ ρ are matrix elements of Q k in the corresponding standard basis. The linear relations given in (17) or a part of the so-called intertwining relations among SDCs together with the unitarity conditions given in (16) are sufficient in solving these SDCs as has been shown in [14] for the corresponding Brauer algebra case. Using (16) and (17)…”

Section: The Birman-wenzl Algebra C F (R Q) In the Non-standard Basimentioning

confidence: 93%

Subduction coefficients of Birman-Wenzl algebras and Racah coefficients of the quantum groupsO_q(n) andSp_q(2m): I. Subduction coefficients

Dai

Pan

Draayer

2001

Irreducible representations of Birman-Wenzl algebras C f (r, q) in the nonstandard basis are discussed. A procedure for evaluating subduction coefficients (SDCs), or the transformation coefficients between standard and non-standard basis of Birman-Wenzl algebras, is formulated based on the linear equation method. SDCs of C f (r, q) with f 4 are derived. Racah coefficients of the quantum groups O q (n) and Sp q (2m) can be obtained from SDCs of Birman-Wenzl algebras C f (r, q) by using the Schur-Weyl-Brauer duality relation between Birman-Wenzl algebras C f (r, q) and the quantum groups O q (n) and Sp q (2m).

“…An alternative orthonormal basis for [λ] is the split basis, denoted by S n − S n1,n2 -basis [9], with n 1 +n 2 = n. By definition, such a basis breaks [λ] (which is, in general, a reducible representation of the direct product subgroup S n1 × S n2 ) in a block-diagonal form:…”

Section: Standard and Split Basismentioning

confidence: 99%

“…The main goal to give explicit and general closed algebraic formulas has not been achieved. Only some special cases have been solved [7,8,9]. There are numerical methods [10,11,12] which are used to approach the issue, but no insight into the structure of the trasformation coefficients can be obtained.…”

Section: Introductionmentioning

confidence: 99%

On the linear equation method for the subduction problem in symmetric groups

Chilla

2006

Abstract. We focus on the tranformation matrices between the standard Young-Yamanouchi basis of an irreducible representation for the symmetric group Sn and the split basis adapted to the direct product subgroups Sn 1 × S n−n 1 . We introduce the concept of subduction graph and we show that it conveniently describes the combinatorial structure of the equation system arisen from the linear equation method. Thus we can outline an improved algorithm to solve the subduction problem in symmetric groups by a graph searching process. We conclude observing that the general matrix form for multiplicity separations, resulting from orthonormalization, can be expressed in terms of Sylvester matrices relative to a suitable inner product in the multiplicity space.