Grothendieck Ring Research Articles

Cluster algebras and quantum affine algebras

Let C be the category of finite-dimensional representations of a quantum affine algebra U q ( g ∧ ) of simply laced type. We introduce certain monoidal subcategories C ℓ ( ℓ ∈ N ) of C , and we study their Grothendieck rings using cluster algebras.

Grothendieck ring of quantum double of finite groups

Let kG be a group algebra, and D(kG) its quantum double. We first prove that the structure of the Grothendieck ring of D(kG) can be induced from the Grothendieck ring of centralizers of representatives of conjugate classes of G. As a special case, we then give an application to the group algebra kD n , where k is a field of characteristic 2 and D n is a dihedral group of order 2n.

Relative Motives and the Theory of Pseudo-finite Fields

We generalize the motivic incarnation morphism from the theory of arithmetic integration to the relative case, where we work over a base variety S over a field k of characteristic zero. We develop a theory of constructible effective Chow motives over S, and we show how to associate a motive to any S-variety. We give a geometric proof of relative quantifier elimination for pseudo-finite fields, and we construct a morphism from the Grothendieck ring of the theory of pseudo-finite fields over S, to the tensor product of ℚ with the Grothendieck ring of constructible effective Chow motives. This morphism yields a motivic realization of parameterized arithmetic integrals. Finally, we define relative arc and jet spaces, and the three relative motivic Poincaré series.

A categorification of quantum [formula omitted

We categorify Lusztig's U ˙ — a version of the quantized enveloping algebra U q ( sl 2 ) . Using a graphical calculus a 2-category U ˙ is constructed whose split Grothendieck ring is isomorphic to the algebra U ˙ . The indecomposable morphisms of this 2-category lift Lusztig's canonical basis, and the Homs between 1-morphisms are graded lifts of a semilinear form defined on U ˙ . Graded lifts of various homomorphisms and antihomomorphisms of U ˙ arise naturally in the context of our graphical calculus. For each positive integer N a representation of U ˙ is constructed using iterated flag varieties that categorifies the irreducible ( N + 1 ) -dimensional representation of U ˙ .

Logarithmic trace and orbifold products

We give a purely equivariant construction of orbifold products for quotient Deligne-Mumford stacks [X/G] where G is an arbitrary linear algebraic group (not necessarily finite). The key to our construction is the definition of the "logarithmic trace" of an equivariant vector bundle. We also prove that there is an orbifold Chern character homomorphism which induces an isomorphism of a canonical summand in the orbifold Grothendieck ring with the orbifold Chow ring. As an application we obtain an associative orbifold product on the Grothendieck ring of [X/G] (as opposed to its inerita stack) taken with complex coefficients.

Ring of subquotients of a finite group II: Pure bisets

This sequel to O. Coşkun (2009) [6] focuses on the structure of the ring Λ ( G ) of subquotients of the finite group G . We show that this ring is isomorphic with the Grothendieck ring of the category of pure ( G , G ) -bisets, which are bisets containing no isogations. We also determine, over a field of characteristic zero, the Mackey functor structure and the primitive idempotents of Λ ( G ) . Main tool of this determination is the marks of subquotients on each other.

Geometric motivic Poincaré series of quasi-ordinary singularities

AbstractThegeometric motivic Poincaré seriesof a germ (S, 0) of complex algebraic variety takes into account the classes in the Grothendieck ring of the jets of arcs through (S, 0). Denef and Loeser proved that this series has a rational form. We give an explicit description of this invariant when (S, 0) is an irreducible germ ofquasi-ordinary hypersurface singularityin terms of the Newton polyhedra of thelogarithmic jacobian ideals. These ideals are determined by thecharacteristic monomialsof a quasi-ordinary branch parametrizing (S, 0).

Periodicities of T-systems and Y-systems

The unrestricted T-system is a family of relations in the Grothendieck ring of the category of the finite-dimensional modules of Yangian or quantum affine algebra associated with a complex simple Lie algebra. The unrestricted T-system admits a reduction called the restricted T-system. In this paper we formulate the periodicity conjecture for the restricted T-systems, which is the counterpart of the known and partially proved periodicity conjecture for the restricted Y-systems. Then, we partially prove the conjecture by various methods: the cluster algebra and cluster category method for the simply laced case, the determinant method for typesAandC, and the direct method for typesA, D, andB(level 2).

Periodicities of T-systems and Y-systems

The unrestricted T-system is a family of relations in the Grothendieck ring of the category of the finite-dimensional modules of Yangian or quantum affine algebra associated with a complex simple Lie algebra. The unrestricted T-system admits a reduction called the restricted T-system. In this paper we formulate the periodicity conjecture for the restricted T-systems, which is the counterpart of the known and partially proved periodicity conjecture for the restricted Y-systems. Then, we partially prove the conjecture by various methods: the cluster algebra and cluster category method for the simply laced case, the determinant method for typesAandC, and the direct method for typesA, D, andB(level 2).

Fusion matrices, generalized Verlinde formulas and partition functions in

The infinite series of logarithmic minimal models is considered in the -extended picture where they are denoted by . As in the rational models, the fusion algebra of is described by a simple graph fusion algebra. The corresponding fusion matrices are mutually commuting, but in general not diagonalizable. Nevertheless, they can be simultaneously brought to Jordan form by a similarity transformation. The spectral decomposition of the fusion matrices is completed by a set of refined similarity matrices converting the fusion matrices into Jordan canonical form consisting of Jordan blocks of rank 1, 2 or 3. The various similarity transformations and Jordan forms are determined from the modular data. This gives rise to a generalized Verlinde formula for the fusion matrices. Its relation to the partition functions in the model is discussed in a general framework. By application of a particular structure matrix and its Moore–Penrose inverse, this Verlinde formula reduces to the generalized Verlinde formula for the associated Grothendieck ring.

Positive motivic measures are counting measures

Let K be a field. A positive motivic measure on the Grothendieck ring K_0(Var_K) is a homomorphism from K_0(Var_K) to the real numbers assigning a nonnegative value to every variety. In this note we show that the only positive motivic measures are the counting measures: measures on K_0(Var_{F_q}) which send a variety to its number of rational points over some fixed finite extension of F_q.

Grothendieck ring and Verlinde-like formula for the -extended logarithmic minimal model

We consider the Grothendieck ring of the fusion algebra of the -extended logarithmic minimal model . Informally, this is the fusion ring of -irreducible characters so it is blind to the Jordan block structures associated with reducible yet indecomposable representations. As in the rational models, the Grothendieck ring is described by a simple graph fusion algebra. The 2p-dimensional matrices of the regular representation are mutually commuting but not diagonalizable. They are brought simultaneously to Jordan form by the modular data coming from the full (3p − 1)-dimensional S-matrix which includes transformations of the p − 1 pseudo-characters. The spectral decomposition yields a Verlinde-like formula that is manifestly independent of the modular parameter τ but is, in fact, equivalent to the Verlinde-like formula recently proposed by Gaberdiel and Runkel involving a τ-dependent S-matrix.

On the classification of the Grothendieck rings of non-self-dual modular categories

We develop a symbolic computational approach to classifying low-rank modular fusion categories, up to finite ambiguity. By a generalized form of Ocneanu rigidity due to Etingof, Ostrik and Nikshych, it is enough to classify modular fusion algebras of a given rank—that is, to determine the possible Grothendieck rings with modular realizations. We use this technique to classify modular categories of rank at most 5 that are non-self-dual, i.e. those for which some object is not isomorphic to its dual object.

A Monoidal Category for Perturbed Defects in Conformal Field Theory

Starting from an abelian rigid braided monoidal category C we define an abelian rigid monoidal category C_F which captures some aspects of perturbed conformal defects in two-dimensional conformal field theory. Namely, for V a rational vertex operator algebra we consider the charge-conjugation CFT constructed from V (the Cardy case). Then C = Rep(V) and an object in C_F corresponds to a conformal defect condition together with a direction of perturbation. We assign to each object in C_F an operator on the space of states of the CFT, the perturbed defect operator, and show that the assignment factors through the Grothendieck ring of C_F. This allows one to find functional relations between perturbed defect operators. Such relations are interesting because they contain information about the integrable structure of the CFT.

Generating series of classes of Hilbert schemes of points on orbifolds

The notion of power structure over the Grothendieck ring of complex quasi-projective varieties is used for describing generating series of classes of Hilbert schemes of zero-dimensional subschemes (“fat points”) on complex orbifolds.

Variations on a question of Larsen and Lunts

Let k k be a field of characteristic zero. Let X X and X ′ X’ be two k k -schemes of finite type having the same class in the Grothendieck ring of varieties. Is it true that X X and X ′ X’ are piecewise isomorphic? This question, originally asked by Larsen and Lunts, and some of its consequences constitute the subject of this article.

Produit eulérien motivique et courbes rationnelles sur les variétés toriques

AbstractWe study the asymptotical behaviour of the moduli space of morphisms of given anticanonical degree from a rational curve to a split toric variety, when the degree goes to infinity. We obtain in this case a geometric analogue of Manin’s conjecture about rational points of bounded height on varieties defined over a global field. The study is led through a generating series whose coefficients lie in a Grothendieck ring of motives, the motivic height zeta function. In order to establish convergence properties of this function, we use a notion of motivic Euler product. It relies on a construction of Denef and Loeser which associates a virtual motive to a first order logic ring formula.

Ring of subquotients of a finite group I: Linearization

We introduce the ring Λ ( G ) of subquotients of a finite group G. As an abelian group, it is free on the set of conjugacy classes of subquotients of the group G. The ring Λ ( G ) integrally extends the Burnside ring and there is a linearization map with range the Grothendieck ring of the Mackey algebra.

Kirillov-Reshetikhin Conjecture: The General Case

We prove the Kirillov–Reshetikhin (KR) conjecture in the general case: for all twisted quantum affine algebras, we prove that the characters of KR modules solve the twisted Q-system [20] and we get explicit formulas for the character of their tensor products (the untwisted case was treated in [16, 33, 34]). The proof provides several new developments for the representation theory of twisted quantum affine algebras, in particular on the twisted Frenkel–Reshetikhin q-characters that we define (expected in [11, 12]) and on the parameterization of simple finite dimensional representations without Drinfeld presentation. We also prove the twisted T-system [30]. As an application, we get a proof of explicit (q)-characters formulas conjectured in various papers. We prove an isomorphism of Grothendieck rings between a twisted quantum affine algebra and the corresponding simply-laced quantum affine algebra.

Differential operators and crystals of extremal weight modules

We give a combinatorial realization of extremal weight crystals over the quantum group of type A + ∞ and their Littlewood–Richardson rule. Based on this description, we show that the Grothendieck ring generated by the isomorphism classes of extremal weight A + ∞ -crystals is isomorphic to the Weyl algebra of infinite rank, and hence each isomorphism class is realized as a differential operator or non-commutative Schur function acting on the algebra of symmetric functions. We also find a duality between extremal weight A + ∞ -crystals and generalized Verma A ∞ -crystals appearing in the crystal of the Fock space with infinite level, which recovers the generalized Cauchy identity for Schur operators in a bijective and crystal theoretic way.