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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 _{K}*C*_{q}(*n*)
to _{K}*C*_{t}(*n*),
where _{K}*C*_{s}(*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 _{K}*C*_{q}(*n*). In case *K*=*Q*
we exhibit explicitly a decomposition of _{Q}*C*_{q}(*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*=*F*_{2} 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
_{F2}*C*_{q}(*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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