The Hopf algebra of skew shapes, torsion sheaves on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi mathvariant="double-struck">A</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">/</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="double-struck">F</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup></mml:math>, and ideals in Hall algebras of monoid

Szczesny, Matt

doi:10.1016/j.aim.2018.03.032

Cited by 13 publications

(13 citation statements)

References 15 publications

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“…We can therefore apply the construction of [50, §4] to produce a module over of the Hall algebra H( A-proj n ). The algebra H( A-proj n ), and its variations, have been studied by Szczesny [40][41][42]. In the setting of the representation theory of quivers over F 1 , which is combinatorial but not split, modules arising from the R • -construction have been studied in [51] where, in particular, a version of Green's theorem is proved.…”

Section: Remark 213mentioning

confidence: 99%

“…From the point of view of algebraic geometry over fields, monoid schemes can be seen as a direct generalization of toric geometry and Kato fans of logarithmic schemes; see [3,6,8,9,27] among others. The central position of monoid schemes within F 1 -geometry is confirmed by their numerous links to other areas of mathematics, such as Weyl groups as algebraic groups over F 1 [29,44], computational methods for toric geometry [7,8,14], a framework for tropical scheme theory [15], applications to representation theory [20,[40][41][42]51] and, last but not least, stable homotopy theory as K-theory over F 1 [3,10], a theme on which we dwell in this paper.…”

Section: Introductionmentioning

confidence: 99%

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Algebraic $$K\!$$-theory and Grothendieck–Witt theory of monoid schemes

2022

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We study the algebraic $$K\!$$ K -theory and Grothendieck–Witt theory of proto-exact categories of vector bundles over monoid schemes. Our main results are the complete description of the algebraic $$K\!$$ K -theory space of an integral monoid scheme X in terms of its Picard group $${{\,\mathrm{Pic}\,}}(X)$$ Pic ( X ) and pointed monoid of regular functions $$\Gamma (X, {\mathcal {O}}_X)$$ Γ ( X , O X ) and a complete description of the Grothendieck–Witt space of X in terms of an additional involution on $${{\,\mathrm{Pic}\,}}(X)$$ Pic ( X ) . We also prove space-level projective bundle formulae in both settings.

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Section: Remark 213mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Algebraic $$K\!$$-theory and Grothendieck–Witt theory of monoid schemes

2022

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“…In this section we consider a subcategory Skew n ⊂ Mod(P n ) F 1 (originally introduced in [10]) consisting of n-dimensional skew shapes. Our goal is to give an explicit description of the product in the ring K sp 0 (Skew n ).…”

Section: Ladder Simple Cyclementioning

confidence: 99%

Split Grothendieck rings of rooted trees and skew shapes via monoid representations

Beers,

Szczesny

2018

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We study commutative ring structures on the integral span of rooted trees and n-dimensional skew shapes. The multiplication in these rings arises from the smash product operation on monoid representations in pointed sets. We interpret these as Grothendieck rings of indecomposable monoid representations over F 1 -the "field" of one element. We also study the base-change homomorphism from t -modules to k[t]-modules for a field k containing all roots of unity, and interpret the result in terms of Jordan decompositions of adjacency matrices of certain graphs.

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“…1 Several interesting "combinatorial" categories are equipped with a proto-exact structure, for instance, the category of matroids [EJS20], the category of representations over a quiver (and more generally any monoid) over "the field with one element" [Szc12], [JS20a], [JS21]. Categories with more algebro-geometric flavors, which are not additive, have been explored in [Szc18], [JS20b], [ELY20].…”

Section: Introductionmentioning

confidence: 99%

Proto-exact categories of modules over semirings and hyperrings

Jun¹,

Szczesny²,

Tolliver³

2022

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Proto-exact categories, introduced by Dyckerhoff and Kapranov, are a generalization of Quillen exact categories which provide a framework for defining algebraic K-theory and Hall algebras in a non-additive setting. This formalism is well-suited to the study of categories whose objects have strong combinatorial flavor.In this paper, we show that the categories of modules over semirings and hyperrings -algebraic structures which have gained prominence in tropical geometry -carry proto-exact structures.In the first part, we prove that the category of modules over a semiring is equipped with a protoexact structure; modules over an idempotent semiring have a strong connection to matroids. We also prove that the category of algebraic lattices L has a proto-exact structure, and furthermore that the subcategory of L consisting of finite lattices is equivalent to the category of finite B-modules as protoexact categories, where B is the Boolean semifield. We also discuss some relations between L and geometric lattices (simple matroids) from this perspective.In the second part, we prove that the category of modules over a hyperring has a proto-exact structure. In the case of finite modules over the Krasner hyperfield K, a well-known relation between finite K-modules and finite incidence geometries yields a combinatorial interpretation of exact sequences.

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The Hopf algebra of skew shapes, torsion sheaves on A/F1n, and ideals in Hall algebras of monoid

Cited by 13 publications

References 15 publications

Algebraic $$K\!$$-theory and Grothendieck–Witt theory of monoid schemes

Algebraic $$K\!$$-theory and Grothendieck–Witt theory of monoid schemes

Split Grothendieck rings of rooted trees and skew shapes via monoid representations

Proto-exact categories of modules over semirings and hyperrings

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