Séminaire Lotharingien de Combinatoire, B41c (1998), 20pp.

Helmut Krämer

Eigenspace Decompositions with Respect to Symmetrized Incidence Mappings

Abstract. Let K denote one of the field of the rationals or the field F(2) with two elements and define H(t,q) to be the K-incidence matrix of the t-sets versus the q-sets of the n-set {1,2,...,n}. This matrix is considered as a linear map of K-vector spaces KCq(n) to KCt(n), where KCs(n) is the K-vector space having the s-sets as a basis. The symmetrized K-incidence matrix (of H(t,q)) is defined to be the symmetric matrix HH(t,q) equal to the transpose of H(t,q) times H(t,q) which is also considered as an endomorphism of KCq(n). In case K=Q we exhibit explicitly a decomposition of QCq(n) into eigenspaces with respect to HH(t,q). A closer examination of the proof of this result yields a canonical decomposition of ker H(t,q) extending work done by Graver and Jurkat.

In case of K=F2 denote HH(q | n):=HH(q-1,q). Then HH(q | n) is a projection hence diagonalizable if n is odd (otherwise nilpotent). In both cases the rank of HH(q | n) is determined; among other results an explicit decomposition of F2Cq(n) into the two eigenspaces with respect to HH(q | n) is obtained provided n is odd.

Received: August 10, 1998; Accepted: January 18, 1999.

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