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

Abstract

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.

A reduced subduction graph and higher multiplicity inSntransformation coefficients

“…In figure 1, the graphical representation of the subduction graph for ( [4,1]; [1], [3,1]), obtained from the overlap of the 2-layer, the 3-layer and the 4-layer, is shown as an example. Note that the 1-layer is not defined for ( [4,1]; [1], [3,1]) due to [λ 1 ] = [1] (see also [12]).…”

“…From subduction graph [12] and by using a suitable Mathematica program [16], we generate the homogeneous linear sistem required to obtain the SDCs. Then we find the kernel of the subduction matrix which provides a non-orthonormalized form for the coefficients.…”

“…Pan and Chen [11] have presented the linear equation method that is particularly useful since it provides q-dependent algebraic solutions for Hecke algebra H n (q), a quantum deformation of the group algebra CS n . In [12] we have given an improved version of such a method which uses the concept of subduction graph to select a minimal set of linear equations solving the subduction problem for symmetric groups in a systematic manner. In this paper we look for a more insight into the structure of the solution for such a system, reducing the number of unknown SDCs and the number of needed equations.…”

“…The layout of the paper is as follows. In the next section, we review some background and we refer the reader to [12], and references therein, for definitions, notations and for more details on the subduction graph method. In section 3, we analyze the structure of the subduction space and we prove two theorems which are useful to reduce the number of unknowns and, consequently, the number of equations for the subduction problem.…”

Selection and identity rules for subductions of type A quantum Iwahori-Hecke algebras

“…Then, we was able to link the freedom in fixing the multiplicity separation to the freedom deriving from the choice of such a Sylvester matrix. This paper represents a natural continuation and one of the possible generalizations of [29]. Here, we focus on the subduction problem involving semisimple representations of type A quantum Iwahori-Hecke algebras and we analize the combinatorial structure of the reduction system arisen in the changing from standard to non-standard bases.…”