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.
An integral second fundamental theorem of invariant theory for partition algebras
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abstract
We prove that the kernel of the action the group algebra of the Weyl group acting on tensor space (via restriction of the action from the general linear group) is a cell ideal with respect to the alternating Murphy basis. This provides an analogue of the second fundamental theory of invariant theory for the partition algebra over an arbitrary commutative ring and proves that the centraliser algebras of the partition algebra are cellular. We also prove similar results for the half partition algebras.
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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.