Enumerative geometry of dormant opers

The purpose of the present paper is to develop the enumerative geometry of dormant $G$-opers for a semisimple algebraic group $G$. In the present paper, we construct a compact moduli stack admitting a perfect obstruction theory by introducing the notion of a dormant faithful twisted $G$-oper (or a "$G$-do'per" for short. Moreover, a semisimple $2$d TQFT (= $2$-dimensional topological quantum field theory) counting the number of $G$-do'pers is obtained by means of the resulting virtual fundamental class. This $2$d TQFT gives an analogue of the Witten-Kontsevich theorem describing the intersection numbers of psi classes on the moduli stack of $G$-do'pers.Comment: 64 pages, the title is changed, some mistakes are correcte

Faithful Squashed Entanglement

Squashed entanglement is a measure for the entanglement of bipartite quantum states. In this paper we present a lower bound for squashed entanglement in terms of a distance to the set of separable states. This implies that squashed entanglement is faithful, that is, strictly positive if and only if the state is entangled. We derive the bound on squashed entanglement from a bound on quantum conditional mutual information, which is used to define squashed entanglement and corresponds to the amount by which strong subadditivity of von Neumann entropy fails to be saturated. Our result therefore sheds light on the structure of states that almost satisfy strong subadditivity with equality. The proof is based on two recent results from quantum information theory: the operational interpretation of the quantum mutual information as the optimal rate for state redistribution and the interpretation of the regularised relative entropy of entanglement as an error exponent in hypothesis testing. The distance to the set of separable states is measured by the one-way LOCC norm, an operationally-motivated norm giving the optimal probability of distinguishing two bipartite quantum states, each shared by two parties, using any protocol formed by local quantum operations and one-directional classical communication between the parties. A similar result for the Frobenius or Euclidean norm follows immediately. The result has two applications in complexity theory. The first is a quasipolynomial-time algorithm solving the weak membership problem for the set of separable states in one-way LOCC or Euclidean norm. The second concerns quantum Merlin-Arthur games. Here we show that multiple provers are not more powerful than a single prover when the verifier is restricted to one-way LOCC operations thereby providing a new characterisation of the complexity class QMA.Comment: 24 pages, 1 figure, 1 table. Due to an error in the published version, claims have been weakened from the LOCC norm to the one-way LOCC nor

The Hecke group algebra of a Coxeter group and its representation theory

Let W be a finite Coxeter group. We define its Hecke-group algebra by gluing together appropriately its group algebra and its 0-Hecke algebra. We describe in detail this algebra (dimension, several bases, conjectural presentation, combinatorial construction of simple and indecomposable projective modules, Cartan map) and give several alternative equivalent definitions (as symmetry preserving operator algebra, as poset algebra, as commutant algebra, ...). In type A, the Hecke-group algebra can be described as the algebra generated simultaneously by the elementary transpositions and the elementary sorting operators acting on permutations. It turns out to be closely related to the monoid algebras of respectively nondecreasing functions and nondecreasing parking functions, the representation theory of which we describe as well. This defines three towers of algebras, and we give explicitly the Grothendieck algebras and coalgebras given respectively by their induction products and their restriction coproducts. This yields some new interpretations of the classical bases of quasi-symmetric and noncommutative symmetric functions as well as some new bases.Comment: v2: 30 pages, 2 figures, extended proof of Prop. 3.25, update of citations, typo and grammar fixes v3: final version: typos and encoding fixe

Highest weight theory for finite-dimensional graded algebras with triangular decomposition

We show that the category of graded modules over a finite-dimensional graded algebra admitting a triangular decomposition can be endowed with the structure of a highest weight category. When the algebra is self-injective, we show furthermore that this highest weight category has tilting modules in the sense of Ringel. This provides a new perspective on the representation theory of such algebras, and leads to several new structures attached to them. There are a wide variety of examples in algebraic Lie theory to which this applies: restricted enveloping algebras, Lusztig's small quantum groups, hyperalgebras, finite quantum groups, and restricted rational Cherednik algebras.Comment: To appear in Adv. Mat

Excluding and constructing of exotic group actions on spheres

Wydział Matematyki i InformatykiPraca dotyczy egzotycznych gładkich działań grup skończonych na sferach z jednym, bądź dwoma punktami stałymi. Pierwszym tematem naszych badań jest wykluczanie gładkich działań grup skończonych na sferach z jednym punktem stałym. Podajemy strategię wykluczania działań z jednym punktem stałym na sferach o zadanym wymiarze. Strategia ta polega na wykorzystaniu własności homologicznych danych dotyczących punktów stałych oraz użycia teorii przecięć. Podajemy nowe algebraiczne warunki, wystarczające do wykluczania działań z jednym punktem stałym. Przedstawiamy algorytm, który pozwala nam wykluczyć rozważane działania. Wspomniany algorytm daje nowe wyniki wykluczające. Praca dotyczy również działań z dwoma punktami stałymi na sferach, dla których struktury modułów grupowych na przestrzeniach stycznych w punktach stałych nie są ze sobą izomorficzne. Pytanie dotyczące takich działań zostało zadane przez Smitha. Hipoteza Laitinena sugeruje negatywną odpowiedź na pytanie Smitha dla grup spełniających określone warunki algebraiczne. Chociaż wspomniana hipoteza nie jest prawdziwa w pełnej ogólności, zachodzi ona jednak dla szeregu grup skończonych. Hipoteza Laitinena pozostaje nierozstrzygnięta dla różnych rodzin grup. Naszym głównym wynikiem tej części rozprawy jest wskazanie nowej nieskończonej rodziny grup skończonych, dla których zachodzi hipoteza Laitinena.The thesis concerns exotic smooth actions of finite groups on spheres with one and two fixed points. The first subject of our research are exclusions of smooth one fixed point actions of finite groups on spheres. We develop a strategy of excluding of such actions on spheres of a given dimension. The strategy relies on homological properties of the fixed point data and intersection theory. We provide new algebraic conditions, sufficient to exclude one fixed point actions. We present an algorithm which allows us to exclude the actions in question. This algorithm provides new exclusion results. This thesis is also concerned with two fixed point actions on spheres having non-isomorphic group module structures on the tangent spaces at the fixed points. The question about the existence of such actions was raised by Smith. There is a conjecture of Laitinen which predicts the negative answer to the Smith question for groups satisfying certain algebraic conditions. Although not true in general, the conjecture holds for many families of finite groups. Still, the Laitinen Conjecture remains unsettled for various families of groups. Our main result of this part is indicating a new infinite family of finite groups for which the Laitinen Conjecture holds

Harmonic analysis on the infinite symmetric group

Let S be the group of finite permutations of the naturals 1,2,... The subject of the paper is harmonic analysis for the Gelfand pair (G,K), where G stands for the product of two copies of S while K is the diagonal subgroup in G. The spherical dual to (G,K) (that is, the set of irreducible spherical unitary representations) is an infinite-dimensional space. For such Gelfand pairs, the conventional scheme of harmonic analysis is not applicable and it has to be suitably modified. We construct a compactification of S called the space of virtual permutations. It is no longer a group but it is still a G-space. On this space, there exists a unique G-invariant probability measure which should be viewed as a true substitute of Haar measure. More generally, we define a 1-parameter family of probability measures on virtual permutations, which are quasi-invariant under the action of G. Using these measures we construct a family {T_z} of unitary representations of G depending on a complex parameter z. We prove that any T_z admits a unique decomposition into a multiplicity free integral of irreducible spherical representations of (G,K). Moreover, the spectral types of different representations (which are defined by measures on the spherical dual) are pairwise disjoint. Our main result concerns the case of integral values of parameter z: then we obtain an explicit decomposition of T_z into irreducibles. The case of nonintegral z is quite different. It was studied by Borodin and Olshanski, see e.g. the survey math.RT/0311369.Comment: AMS Tex, 80 pages, no figure