Polarizations of powers of graded maximal ideals

Abstract

We give a complete combinatorial characterization of all possible polarizations of powers of the graded maximal ideal (x 1 , x 2 , . . . , x m ) n of a polynomial ring in m variables. We also give a combinatorial description of the Alexander duals of such polarizations. In the three variable case m = 3 and also in the power two case n = 2 the descriptions are easily visualized and we show that every polarization defines a (shellable) simplicial ball. We conjecture that any polarization of an Artinian monomial ideal defines a simplicial ball. m and making a minimal generatorof J. We call this the standard polarization.

Polarizations and Hook Partitions

“…Batzies' hypersimplex resolution in particular keeps track of all possible linear syzygies on the monomial minimal generating set of m d , which can be encoded as a graph. In [1], Almousa, Fløystad, and Lohne show that this graph of linear syzygies can be used to characterize all possible polarizations (see Definition 5.1) of m d . We use the aforementioned bijection of basis elements to translate conditions on spanning trees contained within the graph of linear syzygies to rank conditions on submodules generated by elements of an appropriate Schur module.…”

“…We use the aforementioned bijection of basis elements to translate conditions on spanning trees contained within the graph of linear syzygies to rank conditions on submodules generated by elements of an appropriate Schur module. This yields a dictionary between the notation and terminology introduced in [1] and well-established notions arising in the context of Schur modules. Moreover, we extend the results of [1] to polarizations of restricted powers of the maximal ideal, and give an explicit algorithm for checking whether a given graph of linear syzygies induces a well-defined isotone map.…”

“…This yields a dictionary between the notation and terminology introduced in [1] and well-established notions arising in the context of Schur modules. Moreover, we extend the results of [1] to polarizations of restricted powers of the maximal ideal, and give an explicit algorithm for checking whether a given graph of linear syzygies induces a well-defined isotone map. This algorithm opens the door to methods of computing all possible polarizations of m d for any number of variables.…”

“…In Section 5, we summarize the results from [1] giving a complete characterization of all polarizations of powers of the graded maximal ideal. In Section 6, we translate the machinery of Section 5 to the language of hook tableaux (Propositon 6.2).…”

Triangulations of polygons and stacked simplicial complexes: separating their Stanley-Reisner ideals

“…The ideals I(T ) are introduced in [2] where they are shown to be all possible polarizations of the square of the graded maximal ideal (x e ) 2 e∈E in k[x e ] e∈E . If I X is the Stanley-Reisner ring of a stacked simplicial complex we therefore have processes:…”