Representation Ring Research Articles

Computing singularities of the spectra of representation rings of finite groups

Let $G$ be a finite group of order $n$, and $\xi$ an $n$-th primitive root of unity. Consider the affine scheme $C:=\mbox{Spc}({\mathbb Z}[\xi]\otimes_{\mathbb Z} R(G))$ where $R(G)$ is the representation ring of $G$. We study the fibers of the formal tangent sheaf of $C$ by computing their dimension and also finding (and measuring) the singularities of $C$. We present explicit computations for noncommutative groups of small order, and develop practical methods to compute these invariants for an arbitrary finite group.

A filtration on the ring of Laurent polynomials and representations of the general linear Lie algebra

We first present a filtration on the ring Ln of Laurent polynomials such that the direct sum decomposition of its associated graded ring grLn agrees with the direct sum decomposition of grLn, as a module over the complex general linear Lie algebra gl(n), into its simple submodules. Next, generalizing the simple modules occurring in the associated graded ring grLn, we give some explicit constructions of weight multiplicity-free irreducible representations of gl(n).

Representations of a non-pointed Hopf algebra

<abstract><p>In this paper, we construct all the indecomposable modules of a class of non-pointed Hopf algebras, which are quotient Hopf algebras of a class of prime Hopf algebras of GK-dimension one. Then the decomposition formulas of the tensor product of any two indecomposable modules are established. Based on these results, the representation ring of the Hopf algebras is characterized by generators and some relations.</p></abstract>

Lift of fractional D-brane charge to equivariant Cohomotopy theory

The lift of K-theoretic D-brane charge to M-theory was recently hypothesized to land in Cohomotopy cohomology theory. To further check this Hypothesis H, here we explicitly compute the constraints on fractional D-brane charges at ADE-orientifold singularities imposed by the existence of lifts from equivariant K-theory to equivariant Cohomotopy theory, through Boardman’s comparison homomorphism. We check the relevant cases and find that this condition singles out precisely those fractional D-brane charges which do not take irrational values, in any twisted sector. Given that the possibility of irrational D-brane charge has been perceived as a paradox in string theory, we conclude that Hypothesis H serves to resolve this paradox.Concretely, we first explain that the Boardman homomorphism, in the present case, is the map from the Burnside ring to the representation ring of the singularity group given by forming virtual permutation representations. Then we describe an explicit algorithm that computes the image of this comparison map for any finite group. We run this algorithm for binary Platonic groups, hence for finite subgroups of SU(2); and we find explicitly that for the three exceptional subgroups and for the first few cyclic and binary dihedral subgroups the comparison morphism surjects precisely onto the sub-lattice of the real representation ring spanned by the non-irrational characters.

Reconstruction of tensor categories from their structure invariants

In this paper, we study tensor (or monoidal) categories of finite rank over an algebraically closed field $\mathbb F$. Given a tensor category $\mathcal{C}$, we have two structure invariants of $\mathcal{C}$: the Green ring (or the representation ring) $r(\mathcal{C})$ and the Auslander algebra $A(\mathcal{C})$ of $\mathcal{C}$. We show that a Krull-Schmit abelian tensor category $\mathcal{C}$ of finite rank is uniquely determined (up to tensor equivalences) by its two structure invariants and the associated associator system of $\mathcal{C}$. In fact, we can reconstruct the tensor category $\mathcal{C}$ from its two invarinats and the associator system. More general, given a quadruple $(R, A, \phi, a)$ satisfying certain conditions, where $R$ is a $\mathbb{Z}_+$-ring of rank $n$, $A$ is a finite dimensional $\mathbb F$-algebra with a complete set of $n$ primitive orthogonal idempotents, $\phi$ is an algebra map from $A\otimes_{\mathbb F}A$ to an algebra $M(R, A, n)$ constructed from $A$ and $R$, and $a=\{a_{i,j,l}|1< i,j,l<n\}$ is a family of invertible matrices over $A$, we can construct a Krull-Schmidt and abelian tensor category $\mathcal C$ over $\mathbb{F}$ such that $R$ is the Green ring of $\mathcal C$ and $A$ is the Auslander algebra of $\mathcal C$. In this case, $\mathcal C$ has finitely many indecomposable objects (up to isomorphisms) and finite dimensional Hom-spaces. Moreover, we will give a necessary and sufficient condition for such two tensor categories to be tensor equivalent.

Idempotent characters and equivariantly multiplicative splittings of K‐theory

We classify the primitive idempotents of the $p$-local complex representation ring of a finite group $G$ in terms of the cyclic subgroups of order prime to $p$ and show that they all come from idempotents of the Burnside ring. Our results hold without adjoining roots of unity or inverting the order of $G$, thus extending classical structure theorems. We then derive explicit group-theoretic obstructions for tensor induction to be compatible with the resulting idempotent splitting of the representation ring Mackey functor. Our main motivation is an application in homotopy theory: we conclude that the idempotent summands of $G$-equivariant topological $K$-theory and the corresponding summands of the $G$-equivariant sphere spectrum admit exactly the same flavors of equivariant commutative ring structures, made precise in terms of Hill-Hopkins-Ravenel norm maps. This paper is a sequel to the author's earlier work on multiplicative induction for the Burnside ring and the sphere spectrum, see arXiv:1802.01938.

E8 spectral curves

I provide an explicit construction of spectral curves for the affine $\mathrm{E}_8$ relativistic Toda chain. Their closed form expression is obtained by determining the full set of character relations in the representation ring of $\mathrm{E}_8$ for the exterior algebra of the adjoint representation; this is in turn employed to provide an explicit construction of both integrals of motion and the action-angle map for the resulting integrable system. I consider two main areas of applications of these constructions. On the one hand, I consider the resulting family of spectral curves in the context of the correspondences between Toda systems, 5d Seiberg-Witten theory, Gromov-Witten theory of orbifolds of the resolved conifold, and Chern-Simons theory to establish a version of the B-model Gopakumar-Vafa correspondence for the $\mathrm{sl}_N$ L\^e-Murakami-Ohtsuki invariant of the Poincar\'e integral homology sphere to all orders in $1/N$. On the other, I consider a degenerate version of the spectral curves and prove a 1-dimensional Landau-Ginzburg mirror theorem for the Frobenius manifold structure on the space of orbits of the extended affine Weyl group of type $\mathrm{E}_8$ introduced by Dubrovin-Zhang (equivalently, the orbifold quantum cohomology of the type-$\mathrm{E}_8$ polynomial $\mathbb{C} P^1$ orbifold). This leads to closed-form expressions for the flat co-ordinates of the Saito metric, the prepotential, and a higher genus mirror theorem based on the Chekhov-Eynard-Orantin recursion. I will also show how the constructions of the paper lead to a generalisation of a conjecture of Norbury-Scott to ADE $\mathbb{P}^1$-orbifolds, and a mirror of the Dubrovin-Zhang construction for all Weyl groups and choices of marked roots.

Adams operations and symmetries of representation categories

Adams operations are the natural transformations of the representation ring functor on the category of finite groups, and they are one way to describe the usual lambda-ring structure on these rings. From the representation-theoretical point of view, they codify some of the symmetric monoidal structure of the representation category. We show that the monoidal structure on the category alone, regardless of the particular symmetry, determines all the odd Adams operations. On the other hand, we give examples to show that monoidal equivalences do not have to preserve the second Adams operations and to show that monoidal equivalences that preserve the second Adams operations do not have to be symmetric. Along the way, we classify all possible symmetries and all monoidal autoequivalences of representation categories of finite groups.

Green rings of Drinfeld doubles of Taft algebras

In this article, we investigate the representation rings (or Green rings) of the Drinfeld doubles of the Taft algebras. It is shown that these Green rings are commutative rings generated by infinitely many elements subject to certain relations. The generators together with the subjecting relations are given. The stable Green rings of of the Drinfeld doubles of the Taft algebras are also described.

Representation ring of Levi subgroups versus cohomology ring of flag varieties II

For any reductive group G and a parabolic subgroup P with its Levi subgroup L, the first author in [5] introduced a ring homomorphism ξλP:Repλ−polyC(L)→H⁎(G/P,C), where Repλ−polyC(L) is a certain subring of the complexified representation ring of L (depending upon the choice of an irreducible representation V(λ) of G with highest weight λ). In this paper we study this homomorphism for G=Sp(2n) and its maximal parabolic subgroups Pn−k for any 1≤k≤n−1 (with the choice of V(λ) to be the defining representation V(ω1) in C2n). Thus, we obtain a C-algebra homomorphism ξn,k:Repω1−polyC(Sp(2k))→H⁎(IG(n−k,2n),C). Our main result asserts that ξn,k is injective when n tends to ∞ keeping k fixed. Similar results are obtained for the odd orthogonal groups.

A completion theorem for fusion systems

We show that the twisted K-theory of the classifying space of a p-local finite group is isomorphic to the completion of the Grothendieck group of twisted representations of the fusion system with respect to the augmentation ideal of the representation ring of the fusion system. We use this result to compute the K-theory of the Ruiz-Viruel exotic 7-local finite groups.

Dihedral sieving phenomena

Cyclic sieving is a well-known phenomenon where certain interesting polynomials, especially q-analogues, have useful interpretations related to actions and representations of the cyclic group. We propose a definition of sieving for an arbitrary group G and study it for the dihedral group I2(n) of order 2n. This requires understanding the generators of the representation ring of the dihedral group. For n odd, we exhibit several instances of dihedral sieving which involve the generalized Fibonomial coefficients, recently studied by Amdeberhan, Chen, Moll, and Sagan. We also exhibit an instance of dihedral sieving involving Garsia and Haiman’s (q,t)-Catalan numbers.

Motivic concentration theorem

In this short article, given a smooth diagonalizable group scheme G of finite type acting on a smooth quasi-compact quasi-separated scheme X, we prove that (after inverting some elements of representation ring of G) all the information concerning the additive invariants of the quotient stack [X/G] is concentrated in the subscheme of G-fixed points X^G. Moreover, in the particular case where G is connected and the action has finite stabilizers, we compute the additive invariants of [X/G] using solely the subgroups of roots of unity of G. As an application, we establish a Lefschtez-Riemann-Roch formula for the computation of the additive invariants of proper push-forwards.

Volodymyr Vasylovych Kyrychenko

The paper is dedicated to the memory of an outstanding ukrainian mathematician and teacher, one of the founders of Kyiv school of representation theory and ring theory, professor of Taras Shevchenko National University of Kyiv Volodymyr Vasylovych Kyrychenko. Volodymyr Vasylovych was born in 1942 in Penza. He graduated Kyiv University with honors in 1964. He defended his PhD thesis “Representations of hereditary, completely decomposable and Bass orders” under the supervision of D.K. Faddeev. He defended his habilitation “Modules and structural theory of rings” in 1986. He started to work at Kyiv University as an assistant of the Algebra and Mathematical Logic Department in 1967. Since then he served as an Associate Professor and Professor of the Algebra and Mathematical Logic Department. He was the head of the Geometry Department from 1988 till 2016. His main scientific results lay in the field of structural ring theory. In particular, he introduced the notion of the quiver for some special casses of rings and developed the structural theory of hereditary semiserial semidistributive rings. He is the author of about 250 scientific papers and 5 fundamental monographies. In his monographies the theory of semiperfect rings is presented by means of correspond- ing quivers. The monography “Finite dimensional algebras” written jointly with professor Y. Drozd is regarded one of the classical works in algebra and translated from Russian to Chineese, English and Spanish. He was an outstanding teacher and 36 PhD thesis were defended under his supervision. His lectures on analytical geometry and ring theory were always top ranked among students. Professor Kyrychenko was one of the founders of the journal “Algebra and Discrete Mathematics”. He was an editor of a couple of mathematical journals. In 1997 he was one of the founders of the series of international algebraic conferences in Ukraine that were conducted every two years. Since that he was one of the main organizers of each of them.

Specht modules decompose as alternating sums of restrictions of Schur modules

Schur modules give the irreducible polynomial representations of the general linear group G L t \mathrm {GL}_t . Viewing the symmetric group S t \mathfrak {S}_t as a subgroup of G L t \mathrm {GL}_t , we may restrict Schur modules to S t \mathfrak {S}_t and decompose the result into a direct sum of Specht modules, the irreducible representations of S t \mathfrak {S}_t . We give an equivariant Möbius inversion formula that we use to invert this expansion in the representation ring for S t \mathfrak {S}_t for t t large. In addition to explicit formulas in terms of plethysms, we show the coefficients that appear alternate in sign by degree. In particular, this allows us to define a new basis of symmetric functions whose structure constants are stable Kronecker coefficients and which expand with signs alternating by degree into the Schur basis.

A new basis for the representation ring of a Weyl group

Let W W be a Weyl group. In this paper we define a new basis for the Grothendieck group of representations of W W . This basis contains on the one hand the special representations of W W and on the other hand the representations of W W carried by the left cells of W W . We show that the representations in the new basis have a certain bipositivity property.

Multiplicative induction and units for the ring of monomial representations

Let G be a finite group, and let A be a finite abelian G-group. For each subgroup H of G, Ω(H,A) denotes the ring of monomial representations of H with coefficients in A, which is a generalization of the Burnside ring Ω(H) of H. We research the multiplicative induction map Ω(H,A)→Ω(G,A) derived from the tensor induction map Ω(H)→Ω(G), and also research the unit group of Ω(G,A). The results are explained in terms of the first cohomology groups H1(K,A) for K≤G. We see that tensor induction for 1-cocycles plays a crucial role in a description of multiplicative induction. The unit group of Ω(G,A) is identified as a finitely generated abelian group. We especially study the group of torsion units of Ω(G,A), and study the unit group of Ω(G) as well.

Vector bundles over classifying spaces of p‐local finite groups and Benson–Carlson duality

In this paper we obtain a description of the Grothendieck group of complex vector bundles over the classifying space of a p-local finite group in terms of representation rings of subgroups of its Sylow. We also prove a stable elements formula for generalized cohomological invariants of p-local finite groups, which is used to show the existence of unitary embeddings of p-local finite groups. Finally, we show that the augmentation map for the cochains of the classifying space of a p-local finite group is Gorenstein in the sense of Dwyer-Greenlees-Iyengar and obtain some consequences about the cohomology ring of these classifying spaces.

Filtrations of global equivariant K-theory

Arone and Lesh (J Reine Angew Math 604:73–136, 2007; Fund Math 207(1):29–70, 2010) constructed and studied spectrum level filtrations that interpolate between connective (topological or algebraic) K-theory and the Eilenberg–MacLane spectrum for the integers. In this paper we consider (global) equivariant generalizations of these filtrations and another closely related class of filtrations, the modified rank filtrations of the K-theory spectra themselves. We lift Arone and Lesh’s description of the filtration subquotients to the equivariant context and apply it to compute algebraic filtrations on representation rings that arise on equivariant homotopy groups. It turns out that these representation ring filtrations are considerably easier to express in a global equivariant context than over a fixed compact Lie group. Furthermore, they have formal similarities to the filtration on Burnside rings induced by the symmetric products of spheres, which was computed by Schwede (J Am Math Soc 30(3):673–711, 2017).

Iteration of the exterior power on representation rings

In this note we investigate the idea of Michael Atiyah of using, as a possible approach to the Theorem of Feit–Thompson on the solvability of finite groups of odd order, the iterations of the transformation which replaces a representation of a finite group G on a finite dimensional complex vector space E by the difference between the associated representation of G on the sum of exterior powers of E and the trivial representation. We show that G has odd order if and only if the above operation extends to virtual representations and we then express it in terms of the exponential and the Adams operations in the complexified representation ring. We show the relevance of the idea in a concrete example by exhibiting convergence to a non-trivial character.