Grothendieck Ring Research Articles

Burnside rings for Real 2-representation theory: The linear theory

This paper is a fundamental study of the Real [Formula: see text]-representation theory of [Formula: see text]-groups. It also contains many new results in the ordinary (non-Real) case. Our framework relies on a [Formula: see text]-equivariant Morita bicategory, where a novel construction of induction is introduced. We identify the Grothendieck ring of Real [Formula: see text]-representations as a Real variant of the Burnside ring of the fundamental group of the [Formula: see text]-group and study the Real categorical character theory. This paper unifies two previous lines of inquiry, the approach to [Formula: see text]-representation theory via Morita theory and Burnside rings, initiated by the first author and Wendland, and the Real [Formula: see text]-representation theory of [Formula: see text]-groups, as studied by the second author.

On degenerate sections of vector bundles

We consider the locus of sections of a vector bundle on a projective scheme that vanish in higher dimension than expected. We will find the largest components of this locus asymptotically, after applying a high enough twist to the vector bundle. We will also give an interpretation in terms of a limit in the Grothendieck ring of varieties.

Irrationality of motivic zeta functions

Let K0(VarQ)[1/L] denote the Grothendieck ring of Q-varieties with the Lefschetz class inverted. We show that there exists a K3 surface X over Q such that the motivic zeta function ζX(t):=∑n[SymnX]tn regarded as an element in K0(VarQ)[1/L][[t]] is not a rational function in t, thus disproving a conjecture of Denef and Loeser.

Equivariant motivic integration and proof of the integral identity conjecture for regular functions

We develop Denef–Loeser’s motivic integration to an equivariant version and use it to prove the full integral identity conjecture for regular functions. In comparison with Hartmann’s work, the equivariant Grothendieck ring defined in this article is more elementary and it yields the application to the conjecture.

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

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 $\fun$ - the "field" of one element. We also study the base-change homomorphism from $\mt$-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.

How to categorify the ring of integers localized at two

We construct a triangulated monoidal Karoubi closed category with the Grothendieck ring naturally isomorphic to the ring of integers localized at two.

The Structure of the Grothendieck Rings of Wreath Product Deligne Categories and their Generalisations

Abstract Given a tensor category $\mathcal{C}$ over an algebraically closed field of characteristic zero, we may form the wreath product category $\mathcal{W}_n(\mathcal{C})$. It was shown in [10] that the Grothendieck rings of these wreath product categories stabilise in some sense as $n \to \infty $. The resulting “limit” ring, $\mathcal{G}_\infty ^{\mathbb{Z}}(\mathcal{C})$, is isomorphic to the Grothendieck ring of the wreath product Deligne category $S_t(\mathcal{C})$ as defined by [9] (although it is also related to $FI_G$-modules). This ring only depends on the Grothendieck ring $\mathcal{G}(\mathcal{C})$. Given a ring $R$ that is free as a $\mathbb{Z}$-module, we construct a ring $\mathcal{G}_\infty ^{\mathbb{Z}}(R)$ that specialises to $\mathcal{G}_\infty ^{\mathbb{Z}}(\mathcal{C})$ when $R = \mathcal{G}(\mathcal{C})$. We give a description of $\mathcal{G}_\infty ^{\mathbb{Z}}(R)$ using generators very similar to the basic hooks of [5]. We also show that $\mathcal{G}_\infty ^{\mathbb{Z}}(R)$ is a $\lambda $-ring wherever $R$ is and that $\mathcal{G}_\infty ^{\mathbb{Z}}(R)$ is (unconditionally) a Hopf algebra. Finally, we show that $\mathcal{G}_\infty ^{\mathbb{Z}}(R)$ is isomorphic to the Hopf algebra of distributions on the formal neighbourhood of the identity in $(W\otimes _{\mathbb{Z}} R)^\times $, where $W$ is the ring of Big Witt Vectors.

Generalized Orbifold Euler Characteristics on the Grothendieck Ring of Varieties with Actions of Finite Groups

The notion of the orbifold Euler characteristic came from physics at the end of the 1980s. Coincidence (up to sign) of the orbifold Euler characteristics is a necessary condition for crepant resolutions of orbifolds to be mirror symmetric. There were defined higher order versions of the orbifold Euler characteristic and generalized (“motivic”) versions of them. In a previous paper, the authors defined a notion of the Grothendieck ring K 0 fGr ( Var C ) of varieties with actions of finite groups on which the orbifold Euler characteristic and its higher order versions are homomorphisms to the ring of integers. Here, we define the generalized orbifold Euler characteristic and higher order versions of it as ring homomorphisms from K 0 fGr ( Var C ) to the Grothendieck ring K 0 ( Var C ) of complex quasi-projective varieties and give some analogues of the classical Macdonald equations for the generating series of the Euler characteristics of the symmetric products of a space.

A Soergel-like category for complex reflection groups of rank one

We introduce analogues of Soergel bimodules for complex reflection groups of rank one. We give an explicit parametrization of the indecomposable objects of the resulting category and give a presentation of its split Grothendieck ring by generators and relations. This ring turns out to be an extension of the Hecke algebra of the reflection group W and a free module of rank $$|W| (|W|-1)+1$$ over the base ring. We also show that it is a generically semisimple algebra if defined over the complex numbers.

On the K-theory coniveau epimorphism for products of Severi–Brauer varieties

For X a product of Severi–Brauer varieties, we conjecture that if the Chow ring of X is generated by Chern classes, then the canonical epimorphism from the Chow ring of X to the graded ring associated to the coniveau filtration of the Grothendieck ring of X is an isomorphism. We show this conjecture is equivalent to the condition that if G is a split semisimple algebraic group of type AC, B is a Borel subgroup of G and E is a standard generic G-torsor, then the canonical epimorphism from the Chow ring of E∕B to the graded ring associated with the coniveau filtration of the Grothendieck ring of E∕B is an isomorphism. In certain cases we verify this conjecture.

Representations of the small nonstandard quantum groups

In this paper, we study the representations of a class of small nonstandard quantum group over which the isomorphism classes of all indecomposable modules are classified, and the decomposition formulas of the tensor product of arbitrary indecomposable modules and simple (or projective) modules are established. The projective class rings and Grothendieck rings of are also characterized.

Symmetric tensor categories in characteristic 2

We construct and study a nested sequence of finite symmetric tensor categories Vec=C0⊂C1⊂⋯⊂Cn⊂⋯ over a field of characteristic 2 such that C2n are incompressible, i.e., do not admit tensor functors into tensor categories of smaller Frobenius–Perron dimension. This generalizes the category C1 described by Venkatesh [28] and the category C2 defined by Ostrik. The Grothendieck rings of the categories C2n and C2n+1 are both isomorphic to the ring of real cyclotomic integers defined by a primitive 2n+2-th root of unity, On=Z[2cos⁡(π/2n+1)].

Motivic random variables and representation stability II: Hypersurface sections

We prove geometric and cohomological stabilization results for the universal smooth degree d hypersurface section of a fixed smooth projective variety as d goes to infinity. We show that relative configuration spaces of the universal smooth hypersurface section stabilize in the completed Grothendieck ring of varieties, and deduce from this the stabilization of the Hodge Euler characteristic of natural families of local systems constructed from the vanishing cohomology. We prove explicit formulas for the stable values using a probabilistic interpretation, along with the natural analogs in point counting over finite fields. We explain how these results provide new geometric examples of a weak version of representation stability for symmetric, symplectic, and orthogonal groups. This interpretation of representation stability was studied in the prequel [20] for configuration spaces.

Arithmetic of the moduli of semistable elliptic surfaces

We prove a new sharp asymptotic with the lower order term of zeroth order on $\mathcal{Z}_{\mathbb{F}_q(t)}(\mathcal{B})$ for counting the semistable elliptic curves over $\mathbb{F}_q(t)$ by the bounded height of discriminant $\Delta(X)$. The precise count is acquired by considering the moduli of nonsingular semistable elliptic fibrations over $\mathbb{P}^{1}$, also known as semistable elliptic surfaces, with $12n$ nodal singular fibers and a distinguished section. We establish a bijection of $K$-points between the moduli functor of semistable elliptic surfaces and the stack of morphisms $\mathcal{L}_{1,12n} \cong \mathrm{Hom}_n(\mathbb{P}^{1}, \overline{\mathcal{M}}_{1,1})$ where $\overline{\mathcal{M}}_{1,1}$ is the Deligne-Mumford stack of stable elliptic curves and $K$ is any field of characteristic $\neq 2,3$. For $\mathrm{char}(K)=0$, we show that the class of $\mathrm{Hom}_n(\mathbb{P}^1,\mathcal{P}(a,b))$ in the Grothendieck ring of $K$-stacks, where $\mathcal{P}(a,b)$ is a 1-dimensional $(a,b)$ weighted projective stack, is equal to $\mathbb{L}^{(a+b)n+1}-\mathbb{L}^{(a+b)n-1}$. Consequently, we find that the motive of the moduli $\mathcal{L}_{1,12n}$ is $\mathbb{L}^{10n + 1}-\mathbb{L}^{10n - 1}$ and the cardinality of the set of weighted $\mathbb{F}_q$-points to be $\#_q(\mathcal{L}_{1,12n}) = q^{10n + 1}-q^{10n - 1}$. In the end, we formulate an analogous heuristic on $\mathcal{Z}_{\mathbb{Q}}(\mathcal{B})$ for counting the semistable elliptic curves over $\mathbb{Q}$ by the bounded height of discriminant $\Delta$ through the global fields analogy.

The motivic nearby fiber and degeneration of stable rationality

We prove that stable rationality specializes in regular families whose fibers are integral and have at most ordinary double points as singularities. Our proof is based on motivic specialization techniques and the criterion of Larsen and Lunts for stable rationality in the Grothendieck ring of varieties.

Grothendieck Ring of Varieties with Actions of Finite Groups

AbstractWe define a Grothendieck ring of varieties with actions of finite groups and show that the orbifold Euler characteristic and the Euler characteristics of higher orders can be defined as homomorphisms from this ring to the ring of integers. We describe two natural λ-structures on the ring and the corresponding power structures over it and show that one of these power structures is effective. We define a Grothendieck ring of varieties with equivariant vector bundles and show that the generalized (‘motivic’) Euler characteristics of higher orders can be defined as homomorphisms from this ring to the Grothendieck ring of varieties extended by powers of the class of the complex affine line. We give an analogue of the Macdonald type formula for the generating series of the generalized higher-order Euler characteristics of wreath products.

Chow Filtration on Representation Rings of Algebraic Groups

Abstract We introduce and study a filtration on the representation ring $R(G)$ of an affine algebraic group $G$ over a field. This filtration, which we call Chow filtration, is an analogue of the coniveau filtration on the Grothendieck ring of a smooth variety. We compare it with the other known filtrations on $R(G)$ and show that all three define on $R(G)$ the same topology. For any $n\geq 1$, we compute the Chow filtration on $R(G)$ for the special orthogonal group $G:=O^+(2n+1)$. In particular, we show that the graded group associated with the filtration is torsion-free. On the other hand, the Chow ring of the classifying space of $G$ over any field of characteristic $\ne 2$ is known to contain non-zero torsion elements. As a consequence, any sufficiently good approximation of the classifying space yields an example of a smooth quasi-projective variety $X$ such that its Chow ring is generated by Chern classes and at the same time contains non-zero elements vanishing under the canonical homomorphism onto the graded ring associated with the coniveau filtration on the Grothendieck ring of $X$.

Categorical Relations Between Langlands Dual Quantum Affine Algebras: Exceptional Cases

We first compute the denominator formulas for quantum affine algebras of all exceptional types. Then we prove the isomorphisms among Grothendieck rings of categories $C_Q^{(t)}$ $(t=1,2,3)$, $\mathscr{C}_{\mathscr{Q}}^{(1)}$ and $\mathscr{C}_{\mathfrak{Q}}^{(1)}$. These results give Dorey's rule for all exceptional affine types, prove the conjectures of Kashiwara-Kang-Kim and Kashiwara-Oh, and provides the partial answers of Frenkel-Hernandez on Langlands duality for finite dimensional representations of quantum affine algebras of exceptional types.

Quantum Grothendieck ring isomorphisms, cluster algebras and Kazhdan-Lusztig algorithm

We establish ring isomorphisms between quantum Grothendieck rings of certain remarkable monoidal categories CQ,Bn and CQ,A2n−1 of finite-dimensional representations of quantum affine algebras of types Bn(1) and A2n−1(1), respectively. Our proof relies in part on the corresponding quantum cluster algebra structures. Moreover, we prove that our isomorphisms specialize at t=1 to the isomorphisms of (classical) Grothendieck rings obtained recently by Kashiwara, Kim and Oh by other methods. As a consequence, we prove a conjecture formulated by the first author in 2002: the multiplicities of simple modules in standard modules in CQ,Bn are given by the specialization of certain analogues of Kazhdan-Lusztig polynomials and the coefficients of these polynomials are positive.

Decomposition Rules for the Ring of Representations of Non-Archimedean GLn

Abstract Let $\mathcal{R}$ be the Grothendieck ring of complex smooth finite-length representations of the sequence of p-adic groups $\{GL_n(F)\}_{n=0}^\infty $, with multiplication defined through parabolic induction. We study the problem of the decomposition of products of irreducible representations in $\mathcal{R}$. We obtain a necessary condition on irreducible factors of a given product by introducing a width invariant. Width $1$ representations form the previously studied class of ladder representations. We later focus on the case of a product of two ladder representations, for which we establish that all irreducible factors appear with multiplicity one. Finally, we propose a general rule for the composition series of a product of two ladder representations and prove its validity for cases in which the irreducible factors correspond to smooth Schubert varieties.