Hilbert series for twisted commutative algebras

Sam, Steven V; Snowden, Andrew

doi:10.5802/alco.9

Cited by 20 publications

(15 citation statements)

References 14 publications

(9 reference statements)

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“…For example, a finite generation property of such ideals up to symmetry has been interested in algebraic statistics (see [AH, HS] and a survey [Dr]). From representation theory point of view, a study of such ideals can be considered as a special instance of twisted commutative algebra [SS1,SS2] and FI-modules [CEF]. In this paper, motivated by commutative algebra questions posed by Le,Nagel,Nguyen and Römer [LNNR1,LNNR2], we study Betti tables of monomial ideals fixed by permutations of the variables.…”

Section: Introductionmentioning

confidence: 99%

Betti tables of monomial ideals fixed by permutations of the variables

Murai

2020

Trans. Amer. Math. Soc.

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Let S n be a polynomial ring with n variables over a field and {I n } n≥1 a chain of ideals such that each I n is a monomial ideal of S n fixed by permutations of the variables. In this paper, we present a way to determine all nonzero positions of Betti tables of I n for all large intergers n from the Z m -graded Betti table of I m for some integer m. Our main result shows that the projective dimension and the regularity of I n eventually become linear functions on n, confirming a special case of conjectures posed by Le, Nagel, Nguyen and Römer.

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Section: Introductionmentioning

confidence: 99%

Betti tables of monomial ideals fixed by permutations of the variables

Murai

2020

Trans. Amer. Math. Soc.

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“…The simple objects are the representations λ placed in degree |λ| with 0 in all other degrees. For an explicit description of these categories and equivalences see [SS2,1.2] [SS3,].…”

Section: The Category Rep(s ∞ )mentioning

confidence: 99%

Deligne categories and representations of the infinite symmetric group

Barter

Entova-Aizenbud

Heidersdorf

2019

Advances in Mathematics

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We establish a connection between two settings of representation stability for the symmetric groups Sn over C. One is the symmetric monoidal category Rep(S∞) of algebraic representations of the infinite symmetric group S∞ = n Sn, related to the theory of FImodules. The other is the family of rigid symmetric monoidal Deligne categories Rep(St), t ∈ C, together with their abelian versions Rep ab (St), constructed by Comes and Ostrik.We show that for any t ∈ C the natural functor Rep(S∞) → Rep ab (St) is an exact symmetric faithful monoidal functor, and compute its action on the simple representations of S∞.Considering the highest weight structure on Rep ab (St), we show that the image of any object of Rep(S∞) has a filtration with standard objects in Rep ab (St).As a by-product of the proof, we give answers to the questions posed by P. Deligne concerning the cohomology of some complexes in the Deligne category Rep(St), and their specializations at non-negative integers n.

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“…Category of polynomial representations. The category P ol has several equivalent descriptions, see [SS,Section 5], [En].…”

Section: Tensor Product Categorificationmentioning

confidence: 99%