Rook sums in the symmetric group algebra satisfy an explicit product rule, have linearly factorable minimal polynomials, and generate ideals whose sizes count pattern-avoiding permutations.
A note on the Weingarten function
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abstract
The aim of this note is to compare work of Formanek \cite{formanek2} on a certain construction of central polynomials with that of Collins \cite{Coll} on integration on unitary groups. These two quite disjoint topics share the construction of the same function on the symmetric group, which the second author calls {\em Weingarten function}. By joining these two approaches we succeed in giving a simplified and {\em very natural} presentation of both Formanek and Collins's Theory.
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Rook sums in the symmetric group algebra
Rook sums in the symmetric group algebra satisfy an explicit product rule, have linearly factorable minimal polynomials, and generate ideals whose sizes count pattern-avoiding permutations.