Representation Ring Research Articles

Solution to the Clebsch–Gordan problem for string algebras

The category of modules over a string algebra is equipped with a tensor product defined point-wise and arrow-wise in terms of the underlying quiver. In the present article we investigate how this tensor product interacts with the classification of indecomposables. We apply the results obtained to solve the Clebsch–Gordan problem for string algebras. Moreover, we describe the corresponding representation ring and tensor ideals in the module category.

Homology of graded Hecke algebras

Let H be a graded Hecke algebra with complex deformation parameters and Weyl group W . We show that the Hochschild, cyclic and periodic cyclic homologies of H are all independent of the parameters, and compute them explicitly. We use this to prove that, if the deformation parameters are real, the collection of irreducible tempered H -modules with real central character forms a Q -basis of the representation ring of W . Our method involves a new interpretation of the periodic cyclic homology of finite type algebras, in terms of the cohomology of a sheaf over the underlying complex affine variety.

On the Kummer construction

We discuss a generalization of the Kummer construction. Namely an integral representation of a finite group produces an action on an abelian variety and, via a crepant resolution of the quotient, this gives rise to a higher dimensional variety with trivial canonical class and first cohomology. We use virtual Poincare polynomials with coefficients in a ring of representations and McKay correspondence to compute cohomology of such Kummer varieties.

Polynomial invariants for a semisimple and cosemisimple Hopf algebra of finite dimension

We introduce new polynomial invariants of a finite-dimensional semisimple and cosemisimple Hopf algebra A over a field k by using the braiding structures of A . The coefficients of polynomial invariants are integers if k is a finite Galois extension of Q , and A is a scalar extension of some finite-dimensional semisimple Hopf algebra over Q . Furthermore, we show that our polynomial invariants are indeed tensor invariants of the representation category of A , and recognize the difference between the representation category and the representation ring of A . Actually, by computing and comparing polynomial invariants, we find new examples of pairs of Hopf algebras whose representation rings are isomorphic, but whose representation categories are distinct.

Reflection equation algebras, coideal subalgebras, and their centres

Reflection equation algebras and related $${U{_q}(\mathfrak g)}$$ -comodule algebras appear in various constructions of quantum homogeneous spaces and can be obtained via transmutation or equivalently via twisting by a cocycle. In this paper we investigate algebraic and representation theoretic properties of such so called ‘covariantized’ algebras, in particular concerning their centres, invariants, and characters. The locally finite part $${F_l(U{_q} (\mathfrak g))}$$ of $${U{_q}(\mathfrak g)}$$ with respect to the left adjoint action is a special example of a covariantized algebra. Generalising Noumi’s construction of quantum symmetric pairs we define a coideal subalgebra B f of $${U{_q}(\mathfrak g)}$$ for each character f of a covariantized algebra. We show that for any character f of $${F_l(U{_q}(\mathfrak g))}$$ the centre Z(B f ) canonically contains the representation ring $${{\rm Rep}(\mathfrak g)}$$ of the semisimple Lie algebra $${\mathfrak g}$$ . We show moreover that for $${\mathfrak g = {\mathfrak sl}_n(\mathbb C)}$$ such characters can be constructed from any invertible solution of the reflection equation and hence we obtain many new explicit realisations of $${{\rm Rep}({\mathfrak sl}_n(\mathbb C))}$$ inside $${U_q({\mathfrak sl}_n(\mathbb C))}$$ . As an example we discuss the solutions of the reflection equation corresponding to the Grassmannian manifold Gr(m,2m) of m-dimensional subspaces in $${{\mathbb C}^{2m}}$$ .

THE DRINFEL'D DOUBLE AND TWISTING IN STRINGY ORBIFOLD THEORY

This paper exposes the fundamental role that the Drinfel'd double D(k[G]) of the group ring of a finite group G and its twists Dβ(k[G]), β ∈ Z3(G,k*) as defined by Dijkgraaf–Pasquier–Roche play in stringy orbifold theories and their twistings. The results pertain to three different aspects of the theory. First, we show that G-Frobenius algebras arising in global orbifold cohomology or K-theory are most naturally defined as elements in the braided category of D(k[G])-modules. Secondly, we obtain a geometric realization of the Drinfel'd double as the global orbifold K-theory of global quotient given by the inertia variety of a point with a G action on the one hand and more stunningly a geometric realization of its representation ring in the braided category sense as the full K-theory of the stack [pt/G]. Finally, we show how one can use the co-cycles β above to twist the global orbifold K-theory of the inertia of a global quotient and more importantly, the stacky K-theory of a global quotient [X/G]. This corresponds to twistings with a special type of two-gerbe.

On the representation ring of the polynomial algebra over a perfect field

We consider the tensor product of modules over the polynomial algebra corresponding to the usual tensor product of linear operators. We present a general description of the representation ring in case the ground field k is perfect. It is made explicit in the special cases when k is real closed respectively algebraically closed. Furthermore, we discuss the generalisation of this problem to representations of quivers. In particular the representation ring of quivers of extended Dynkin type A is provided.

On the Structure of the Fusion Ideal

We prove that there is a finite level-independent bound on the number of relations defining the fusion ring of positive energy representations of the loop group of a simple, simply connected Lie group. As an illustration, we compute the fusion ring of $G_2$ at all levels.

The Bott cofiber sequence in deformation K-theory and simultaneous similarity in U(n)

AbstractWe show that there is a homotopy cofiber sequence of spectra relating Carlsson's deformation K-theory of a group G to its “deformation representation ring,” analogous to the Bott periodicity sequence relating connective K-theory to ordinary homology. We then apply this to study simultaneous similarity of unitary matrices.

On the Representation Ring of a Quiver

The Clebsch–Gordan problem for quiver representations is the problem of decomposing the tensor product of any two representations of a quiver, where the tensor product is defined point-wise and arrow-wise. We introduce the so-called characteristic representations and decompose their tensor product. This is then applied to solve the Clebsch–Gordan problem for quivers of type $\mathbb{A}_n$ and $\mathbb{D}_n$ . In both cases we also provide an explicit description of the representation ring.

The Crossed Burnside Ring, the Drinfel’d Double, and the Dade Group of a p-Group

Let p be a prime number. This paper solves the question of the difference between the rank of the crossed Burnside ring B c (P) of a finite p-group P and of the rational representation ring \(R_{{\mathbb{Q}}} (\mathcal{D}(P))\) of the Drinfel’d double \(\mathcal{D}(P)\) of the group algebra ℚ P. The difference is represented by using the Dade groups of certain subgroups of P.

BRAIDING AND ENTANGLEMENT IN SPIN NETWORKS: A COMBINATORIAL APPROACH TO TOPOLOGICAL PHASES

The spin network quantum simulator relies on the su(2) representation ring (or its q-deformed counterpart at q = root of unity) and its basic features naturally include (multipartite) entanglement and braiding. In particular, q-deformed spin network automata are able to perform efficiently approximate calculations of topological invarians of knots and 3-manifolds. The same algebraic background is shared by 2D lattice models supporting topological phases of matter that have recently gained much interest in condensed matter physics. These developments are motivated by the possibility to store quantum information fault-tolerantly in a physical system supporting fractional statistics since a part of the associated Hilbert space is insensitive to local perturbations. Most of currently addressed approaches are framed within a "double" quantum Chern–Simons field theory, whose quantum amplitudes represent evolution histories of local lattice degrees of freedom.We propose here a novel combinatorial approach based on "state sum" models of the Turaev–Viro type associated with SU(2)q-colored triangulations of the ambient 3-manifolds. We argue that boundary 2D lattice models (as well as observables in the form of colored graphs satisfying braiding relations) could be consistently addressed. This is supported by the proof that the Hamiltonian of the Levin–Wen condensed string net model in a surface Σ coincides with the corresponding Turaev–Viro amplitude on Σ × [0,1] presented in the last section.

Yang–Mills theory over surfaces and the Atiyah–Segal theorem

In this paper we explain how Morse theory for the Yang-Mills functional can be used to prove an analogue, for surface groups, of the Atiyah-Segal theorem. Classically, the Atiyah-Segal theorem relates the representation ring R(\Gamma) of a compact Lie group $\Gamma$ to the complex K-theory of the classifying space $B\Gamma$. For infinite discrete groups, it is necessary to take into account deformations of representations, and with this in mind we replace the representation ring by Carlsson's deformation $K$--theory spectrum $\K (\Gamma)$ (the homotopy-theoretical analogue of $R(\Gamma)$). Our main theorem provides an isomorphism in homotopy $\K_*(\pi_1 \Sigma)\isom K^{-*}(\Sigma)$ for all compact, aspherical surfaces $\Sigma$ and all $*>0$. Combining this result with work of Tyler Lawson, we obtain homotopy theoretical information about the stable moduli space of flat unitary connections over surfaces.

Some remarks on the ring of twisted tilting modules for algebraic groups

In a recent paper [S. Doty, A. Henke, Decomposition of tensor products of modular irreducibles for SL 2, Q. J. Math. 56 (2005) 189–207], Doty and Henke give a decomposition of the tensor product of two rational simple modules for the special linear group of degree 2 over an algebraically closed field of characteristic p > 0 . In performing this calculation it proved useful to know that the simple modules are twisted tensor products of tilting modules. It seems natural therefore to consider the ring of twisted tilting modules for a semisimple group G (a subring of the representation ring of G). However, we quickly specialize to the case in which G is the special linear group of degree 2. We show that (in this case) the ring is reduced and describe associated varieties. We give formulas from which one may determine the multiplicities of the indecomposable module summands of the tensor product of twisted tilting modules.

Algebraic cycles and completions of equivariant K-theory

Let G be a complex, linear algebraic group acting on an algebraic space X. The purpose of this article is to prove a Riemann-Roch theorem (Theorem 6.5) that gives a description of the completion of the equivariant Grothendieck group G0(G,X)⊗C at any maximal ideal of the representation ring R(G)⊗C in terms of equivariant cycles. The main new technique for proving this theorem is our nonabelian completion theorem (Theorem 5.3) for equivariant K-theory. Theorem 5.3 generalizes the classical localization theorems for diagonalizable group actions to arbitrary groups

The rank of a quiver representation

We define a functor which gives the “global rank of a quiver representation” and prove that it has nice properties which make it a generalization of the rank of a linear map. We demonstrate how to construct other “rank functors” for a quiver Q, which induce ring homomorphisms (called “rank functions”) from the representation ring of Q to Z . These rank functions give discrete numerical invariants of quiver representations, useful for computing tensor product multiplicities of representations and determining some structure of the representation ring. We also show that in characteristic 0, rank functors commute with the Schur operations on quiver representations, and the homomorphisms induced by rank functors are λ-ring homomorphisms.

On Inverting the Koszul Complex

Let V be an n-dimensional vector space. We give a direct construction of an exact sequence that gives a GL(V)-equivariant “resolution” of each symmetric power S t V in terms of direct sums of tensor products of the form ∧ i 1 V ⊗···⊗ ∧ i p V. This exact sequence corresponds to inverting the relation in the representation ring of GL(V) that is described by the Koszul complex, and has appeared before in work by B. Totaro, analogously to a construction of K. Akin involving the normalized bar resolution. Our approach yields a concrete description of the differentials, and provides an alternate direct proof that .

Quantum algebras with representation ring of % MathType!MTEF!2!1!+- % feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj % xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab-Xc8Zjab-vc8Snaa % BaaaleaacGaS0HOmaaqabaaaaa!46DC! $$ \mathfrak{s}\mathfrak{l}_2 $$ type

A reducible representation of the Temperley-Lieb algebra is constructed on a tensor product of n-dimensional spaces. As a centralizer of this action, we obtain a quantum algebra (quasi-triangular Hopf algebra) with the representation ring that is equivalent to the representation ring of the \( \mathfrak{s}\mathfrak{l}_2 \) Lie algebra. Bibliography: 23 titles.

The forgetful map in rational K-theory

Let G be a connected reductive algebraic group acting on a scheme X. Let R(G) denote the representation ring of G, and let I be the ideal in R(G) of virtual representations of rank 0. Let G(X) (resp. G(G,X)) denote the Grothendieck group of coherent sheaves (resp. G-equivariant coherent sheaves) on X. Merkurjev proved that if the fundamental group of G is torsion-free, then the map of G(G,X)/IG(G,X) to G(X) is an isomorphism. Although this map need not be an isomorphism if the fundamental group of G has torsion, we prove that without the assumption on the fundamental group of G, this map is an isomorphism after tensoring with the rational numbers.

Generalized Artin and Brauer induction for compact Lie groups

Let G be a compact Lie group. We present two induction theorems for certain generalized G-equivariant cohomology theories. The theory applies to G-equivariant K-theory K G , and to the Borel cohomology associated with any complex oriented cohomology theory. The coefficient ring of K G is the representation ring R(G) of G. When G is a finite group the induction theorems for K G coincide with the classical Artin and Brauer induction theorems for R(G).