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Stanley decompositions of the bracket ring. (English) Zbl 0727.13005
We give an explicit Stanley decomposition of the bracket ring 𝔹 n,d , that is, the commutative ring generated by the d×d-minors of a generic n×d-matrix. A Stanley decomposition is a direct sum decomposition of the additive group of the ring, each summand of which is a bracket monomial times a subring generated freely by brackets. The decomposition is obtained via a shelling of the simplicial complex determined by the standard tableaux. Our construction has important applications in the Cushman-Sanders normal form theory for nilpotent vector fields.
MSC:
13C40Linkage, complete intersections and determinantal ideals
13F20Polynomial rings and ideals
14M12Determinantal varieties