Grothendieck Ring Research Articles

Cohomological and motivic inclusion–exclusion

We categorify the inclusion–exclusion principle for partially ordered topological spaces and schemes to a filtration on the derived category of sheaves. As a consequence, we obtain functorial spectral sequences that generalize the two spectral sequences of a stratified space and certain Vassiliev-type spectral sequences; we also obtain Euler characteristic analogs in the Grothendieck ring of varieties. As an application, we give an algebro-geometric proof of Vakil and Wood's homological stability conjecture for the space of smooth hypersurface sections of a smooth projective variety. In characteristic zero this conjecture was previously established by Aumonier via topological methods.

Representations of shifted quantum affine algebras and cluster algebras I: The simply laced case

AbstractWe introduce a family of cluster algebras of infinite rank associated with root systems of type , , . We show that suitable completions of these cluster algebras are isomorphic to the Grothendieck rings of the categories of the corresponding shifted quantum affine algebras. The cluster variables of a class of distinguished initial seeds are certain formal power series defined by E. Frenkel and the second author, which satisfy a system of functional relations called ‐system. We conjecture that all cluster monomials are classes of simple objects of . In the final section, we show that these cluster algebras contain infinitely many cluster subalgebras isomorphic to the coordinate ring of the open double Bruhat cell of the corresponding simple simply connected algebraic group. This explains the similarity between ‐system relations and certain generalized minor identities discovered by Fomin and Zelevinsky.

Low-degree Hurwitz stacks in the Grothendieck ring

For $2 \leq d \leq 5$, we show that the class of the Hurwitz space of smooth degree $d$, genus $g$ covers of $\mathbb {P}^1$ stabilizes in the Grothendieck ring of stacks as $g \to \infty$, and we give a formula for the limit. We also verify this stabilization when one imposes ramification conditions on the covers, and obtain a particularly simple answer for this limit when one restricts to simply branched covers.

Quantization of Virtual Grothendieck Rings and Their Structure Including Quantum Cluster Algebras

Quantization of Virtual Grothendieck Rings and Their Structure Including Quantum Cluster Algebras

Quotient singularities in the Grothendieck ring of varieties

Let G G be a finite group, X X be a smooth complex projective variety with a faithful G G -action, and Y Y be a resolution of singularities of X / G X/G . Larsen and Lunts asked whether [ X / G ] − [ Y ] [X/G]-[Y] is divisible by [ A 1 ] [\mathbb {A}^1] in the Grothendieck ring of varieties. We show that the answer is negative if B G BG is not stably rational and affirmative if G G is abelian. The case when X = Z n X=Z^n for some smooth projective variety Z Z and G = S n G=S_n acts by permutation of the factors is of particular interest. We make progress on it by showing that [ Z n / S n ] − [ Z ⟨ n ⟩ / S n ] [Z^n/S_n]-[Z\langle n\rangle / S_n] is divisible by [ A 1 ] [\mathbb {A}^1] , where Z ⟨ n ⟩ Z\langle n\rangle is Ulyanov’s polydiagonal compactification of the n n th configuration space of Z Z .

Strong Duality Data of Type A and Extended T-Systems

AbstractThe extended T-systems are a number of relations in the Grothendieck ring of the category of finite-dimensional modules over the quantum affine algebras of types $$A_n^{(1)}$$ A n ( 1 ) and $$B_n^{(1)}$$ B n ( 1 ) , introduced by Mukhin and Young as a generalization of the T-systems. In this paper we establish the extended T-systems for more general modules, which are constructed from an arbitrary strong duality datum of type A. Our approach does not use the theory of q-characters, and so also provides a new proof to the original Mukhin–Young’s extended T-systems.

Characters of [formula omitted] and vertex operators

In this paper, we present a vertex operator approach to construct and compute all complex irreducible characters of the general linear group GLn(Fq). Green's theory of GLn(Fq) is recovered and enhanced under the realization of the Grothendieck ring of representations RG=⨁n≥0R(GLn(Fq)) as two isomorphic Fock spaces associated to two infinite-dimensional F-equivariant Heisenberg Lie algebras hˆF‾ˆq and hˆF‾q, where F is the Frobenius automorphism of the algebraically closed field F‾q. Under this picture, the irreducible characters are realized by the Bernstein vertex operators for Schur functions, the characteristic functions of the conjugacy classes are realized by the vertex operators for the Hall-Littlewood functions, and the character table is completely given by matrix coefficients of vertex operators of these two types. One of the features of the current approach is a simpler identification of the Fock space RG as the Hall algebra of symmetric functions via vertex operator calculus, and another is that we are able to compute in general the character table, where Green's degree formula is demonstrated as an example.

On the motivic Higman conjecture

The G-representation variety RG(Σg) parametrizes the representations of the fundamental group π1(Σg) of a closed surface into an algebraic group G. For G the groups of n×n upper triangular matrices or unipotent upper triangular matrices, we compare two methods for computing algebraic invariants of RG(Σg). Using the geometric method initiated by González-Prieto, Logares and Muñoz, based on a Topological Quantum Field Theory (TQFT), we compute the virtual classes of RG(Σg) in the Grothendieck ring of varieties for n=1,…,5. Introducing the notion of algebraic representatives we are able to efficiently compute the TQFT. Using the arithmetic method initiated by Hausel and Rodriguez-Villegas, we compute the E-polynomials of RG(Σg) for n=1,…,10. For both methods, we describe how the computations can be performed algorithmically. Furthermore, we discuss how the representation varieties for the groups of unipotent upper triangular matrices are related to Higman's conjecture. The computations of this paper can be seen as positive evidence towards a generalized motivic version of the conjecture.

New counterexamples to the birational Torelli theorem for Calabi–Yau manifolds

We produce counterexamples to the birational Torelli theorem for Calabi–Yau manifolds in arbitrarily high dimension: this is done by exhibiting a series of non-birational pairs of Calabi–Yau ( n 2 − 1 ) (n^2-1) -folds which, for n ≥ 2 n \geq 2 even, admit an isometry between their middle cohomologies. These varieties also satisfy an L \mathbb {L} -equivalence relation in the Grothendieck ring of varieties, i.e. the difference of their classes annihilates a power of the class of the affine line. We state this last property for a broader class of Calabi–Yau pairs, namely all those which are realized as pushforwards of a general ( 1 , 1 ) (1,1) -section on a homogeneous roof in the sense of Kanemitsu, along its two extremal contractions.

Equivariant K$K$‐theory of flag Bott manifolds of general Lie type

AbstractThe aim of this paper is to describe the equivariant and ordinary Grothendieck ring and the equivariant and ordinary topological ‐ring of flag Bott manifolds of the general Lie type. This will generalize the results on the equivariant and ordinary cohomology of flag Bott manifolds of the general Lie type due to Kaji, Kuroki, Lee, and Suh.

The degenerate Heisenberg category and its Grothendieck ring

The degenerate Heisenberg category and its Grothendieck ring

Multiplicities and Dimensions in Enveloping Tensor Categories

Abstract In a previous paper, semisimple tensor categories were constructed from certain regular Mal’cev categories. In this paper, we calculate the tensor product multiplicities and the categorical dimensions of the simple objects. This yields also the Grothendieck ring. The main tool is the subquotient decomposition of the generating objects.

Grothendieck rings of towers of generalized Weyl algebras in the finite orbit case

Previously we showed that the tensor product of a weight module over a generalized Weyl algebra (GWA) with a weight module over another GWA is a weight module over a third GWA. In this paper we compute tensor products of simple and indecomposable weight modules over generalized Weyl algebras supported on a finite orbit. This allows us to give a complete presentation by generators and relations of the Grothendieck ring of the categories of weight modules over a tower of generalized Weyl algebras in this setting. We also obtain partial results about the split Grothendieck ring. We described the case of infinite orbits in previous work.

Factorization centers in dimension 2 and the Grothendieck ring of varieties

Factorization centers in dimension 2 and the Grothendieck ring of varieties

Higher order Kirillov--Reshetikhin modules for 𝐔 q (A n (1)), imaginary modules and monoidal categorification

Abstract We study the family of irreducible modules for quantum affine 𝔰 ⁢ 𝔩 n + 1 {\mathfrak{sl}_{n+1}} whose Drinfeld polynomials are supported on just one node of the Dynkin diagram. We identify all the prime modules in this family and prove a unique factorization theorem. The Drinfeld polynomials of the prime modules encode information coming from the points of reducibility of tensor products of the fundamental modules associated to A m {A_{m}} with m ≤ n {m\leq n} . These prime modules are a special class of the snake modules studied by Mukhin and Young. We relate our modules to the work of Hernandez and Leclerc and define generalizations of the category 𝒞 - {\mathscr{C}^{-}} . This leads naturally to the notion of an inflation of the corresponding Grothendieck ring. In the last section we show that the tensor product of a (higher order) Kirillov–Reshetikhin module with its dual always contains an imaginary module in its Jordan–Hölder series and give an explicit formula for its Drinfeld polynomial. Together with the results of [D. Hernandez and B. Leclerc, A cluster algebra approach to q-characters of Kirillov–Reshetikhin modules, J. Eur. Math. Soc. (JEMS) 18 2016, 5, 1113–1159] this gives examples of a product of cluster variables which are not in the span of cluster monomials. We also discuss the connection of our work with the examples arising from the work of [E. Lapid and A. Mínguez, Geometric conditions for □ \square -irreducibility of certain representations of the general linear group over a non-archimedean local field, Adv. Math. 339 2018, 113–190]. Finally, we use our methods to give a family of imaginary modules in type D 4 {D_{4}} which do not arise from an embedding of A r {A_{r}} with r ≤ 3 {r\leq 3} in D 4 {D_{4}} .

NON-ISOMORPHIC SMOOTH COMPACTIFICATIONS OF THE MODULI SPACE OF CUBIC SURFACES

Abstract The well-studied moduli space of complex cubic surfaces has three different, but isomorphic, compact realizations: as a GIT quotient ${\mathcal {M}}^{\operatorname {GIT}}$ , as a Baily–Borel compactification of a ball quotient ${(\mathcal {B}_4/\Gamma )^*}$ , and as a compactified K-moduli space. From all three perspectives, there is a unique boundary point corresponding to non-stable surfaces. From the GIT point of view, to deal with this point, it is natural to consider the Kirwan blowup ${\mathcal {M}}^{\operatorname {K}}\rightarrow {\mathcal {M}}^{\operatorname {GIT}}$ , whereas from the ball quotient point of view, it is natural to consider the toroidal compactification ${\overline {\mathcal {B}_4/\Gamma }}\rightarrow {(\mathcal {B}_4/\Gamma )^*}$ . The spaces ${\mathcal {M}}^{\operatorname {K}}$ and ${\overline {\mathcal {B}_4/\Gamma }}$ have the same cohomology, and it is therefore natural to ask whether they are isomorphic. Here, we show that this is in fact not the case. Indeed, we show the more refined statement that ${\mathcal {M}}^{\operatorname {K}}$ and ${\overline {\mathcal {B}_4/\Gamma }}$ are equivalent in the Grothendieck ring, but not K-equivalent. Along the way, we establish a number of results and techniques for dealing with singularities and canonical classes of Kirwan blowups and toroidal compactifications of ball quotients.

A functor for constructing R$R$‐matrices in the category O$\mathcal {O}$ of Borel quantum loop algebras

AbstractWe tackle the problem of constructing ‐matrices for the category associated to the Borel subalgebra of an arbitrary untwisted quantum loop algebra . For this, we define an invertible exact functor from the category linked to to the one linked to . This functor is compatible with tensor products, preserves irreducibility, and interchanges the subcategories and of Hernandez and Leclerc (Algebra Number Theory 10 (2016) 2015–2052). We construct ‐matrices for by applying on the braidings already found for by Hernandez (Represent. Theory 26 (2022) 179–210). We also use the factorization of the latter intertwiners in terms of stable maps to deduce an analogous factorization for our new braidings. We finally obtain as byproducts new relations for the Grothendieck ring as well as a functorial interpretation of a remarkable ring isomorphism of Hernandez–Leclerc.

Multiparameter colored partition category and the product of the reduced Kronecker coefficients

We introduce and study a multiparameter colored partition category CPar(x) by extending the construction of the partition category, over an algebraically closed field k of characteristic zero and for a multiparameter x∈kr. The morphism spaces in CPar(x) have bases in terms of partition diagrams whose parts are colored by elements of the multiplicative cyclic group Cr. We show that the endomorphism spaces of CPar(x) and additive Karoubi envelope of CPar(x) are generically semisimple. The category CPar(x) is rigid symmetric strict monoidal and we give a presentation of CPar(x) as a monoidal category. The path algebra of CPar(x) admits a triangular decomposition with Cartan subalgebra being equal to the direct sum of the group algebras of complex reflection groups G(r,n). We compute the structure constants for the classes of simple modules in the split Grothendieck ring of the category of modules over the path algebra of the downward partition subcategory of CPar(x) in two ways. Among other things, this gives a formula for the product of the reduced Kronecker coefficients in terms of the Littlewood–Richardson coefficients for G(r,n) and certain Kronecker coefficients for the wreath product (Cr×Cr)≀Sn. For r=1, this formula reduces to a formula for the reduced Kronecker coefficients given by Littlewood. We also give two analogues of the Robinson–Schensted correspondence for colored partition diagrams and, as an application, we classify the equivalence classes of Green's left, right and two-sided relations for the colored partition monoid in terms of these correspondences.

On the Grothendieck Ring of the Sequence of Walled Brauer Algebras

In this paper, we provide a diagrammatic approach to study the branching rules for cell modules on a sequence of walled Brauer algebras. This approach also allows us to calculate the structure constants of multiplication over the Grothendieck ring of the sequence.

K-theory of flag Bott manifolds

Abstract The aim of this paper is to describe the topological K-ring in terms of generators and relations of a flag Bott manifold. We apply our results to give a presentation for the topological K-ring, and hence the Grothendieck ring of algebraic vector bundles over flag Bott–Samelson varieties.