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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