In this paper we classify degenerate Verma modules over the linearly compact Lie superalgebra E(4,4). This completes the description of Verma modules over the exceptional linearly compact Lie superalgebras. As in the other cases all degenerate modules and morphisms between them give rise to infinite bilateral complexes which may be viewed as a generalization of de Rham complexes.

In this work, we settle a problem that dates back to the early 2000s. We provide a complete characterization of special matchings of arbitrary Bruhat intervals in Coxeter groups of type A and apply this result to prove a conjecture of Brenti from 2003 concerning the computation of Kazhdan-Lusztig R-polynomials via special matchings. This yields new evidence in support of the Combinatorial Invariance Conjecture.

We study the embeddings of the exceptional infinite-dimensional Lie superalgebra E(1, 6) in the exceptional Lie superalgebras E(5, 10) and E(4, 4). These questions arose in recent works on enhanced symmetries in some supersymmetric theories by N. Garner, S. Raghavendran, I. Saberi and B. Williams.

We construct a duality functor in the category of continuous representations of the Lie superalgebra E(4,4), the only exceptional simple linearly compact Lie superalgebra, for which it wasn't known. This is achieved by constructing a Lie conformal superalgebra of type (4,4), for which E(4,4) is the annihilation algebra. Along the way we obtain an explicit realization of E(4,4) by vector fields on a (4|4)-dimensional supermanifold.

We classify finite irreducible modules over the conformal superalgebra K4′ by their correspondence with finite conformal modules over the associated annihilation superalgebra A(K4′). This is achieved by a complete classification of singular vectors in generalized Verma modules for A(K4′). We also show that morphisms between generalized Verma modules can be arranged in infinitely many bilateral complexes.

We construct a duality functor on the category of continuous representations of linearly compact Lie superalgebras, using representation theory of Lie conformal superalgebras. We compute the dual representations of the generalized Verma modules.

Given a Lie superalgebra $${\mathfrak {g}}$$ with a subalgebra $${\mathfrak {g}}_{\ge 0}$$ , and a finite-dimensional irreducible $${\mathfrak {g}}_{\ge 0}$$ -module F, the induced $${\mathfrak {g}}$$ -module $$M(F)={\mathcal {U}}({\mathfrak {g}})\otimes _{{\mathcal {U}}({\mathfrak {g}}_{\ge 0})}F$$ is called a finite Verma module. In the present paper we classify the non-irreducible finite Verma modules over the largest exceptional linearly compact Lie superalgebra $${\mathfrak {g}}=E(5,10)$$ with the subalgebra $${\mathfrak {g}}_{\ge 0}$$ of minimal codimension. This is done via classification of all singular vectors in the modules M(F). Besides known singular vectors of degree 1,2,3,4 and 5, we discover two new singular vectors, of degrees 7 and 11. We show that the corresponding morphisms of finite Verma modules of degree 1,4,7, and 11 can be arranged in an infinite number of bilateral infinite complexes, which may be viewed as “exceptional” de Rham complexes for E(5, 10).

We provide a weaker version of the generalized lifting property that holds in complete generality for all Coxeter groups, and we use it to show that every parabolic Bruhat interval of a finite Coxeter group is a Coxeter matroid. We also describe some combinatorial properties of the associated polytopes.

We show that a symmetry property that we call the up-down symmetry implies that the Kazhdan–Lusztig Rx-polynomials of a pircon P are a P-kernel, and we show that this property holds in the classical cases. Then, we enhance and extend to this context a duality of Deodhar in parabolic Kazhdan–Lusztig theory.

In this paper we face the study of the representations of the exceptional Lie superalgebra E(5,10). We recall the construction of generalized Verma modules and give a combinatorial description of the restriction to sl_5 of the Verma module induced by the trivial representation. We use this description to classify morphisms between Verma modules of degree one, two and three proving in these cases a conjecture given by Rudakov. A key tool is the notion of dual morphism between Verma modules.

We give a simple characterization of special matchings in lower Bruhat intervals (that is, intervals starting from the identity element) of a Coxeter group. As a byproduct, we obtain some results on the action of special matchings.

In Tsukerman and Williams (Adv Math 285: 766–810, 2015), it is shown that every Bruhat interval of the symmetric group satisfies the so-called generalized lifting property. In this paper, we show that a Coxeter group satisfies this property if and only if it is finite and simply-laced.

We give a new characterization of the peak subalgebra of the algebra of quasisymmetric functions and use this to construct a new basis for this subalgebra. As an application of these results we obtain a combinatorial formula for the Kazhdan–Lusztig polynomials which holds in complete generality and is simpler and more explicit than any existing one. We point out that, in a certain sense, this formula cannot be simplified.

Motivated by the study of the invariant theory of some finite groups, we introduce and study the notion of partially colored labeled forest. A flag-major index is defined on these forests and we study the distribution of this statistic on all partially colored labeled forests and on linear extensions of a fixed partially colored labeled forest. The main results that we obtain are formulas for such distributions which have a very simple factorization form and generalize and unify several known results present in the literature.

We investigate the generalized involution models of the projective reflection groups G(r, p, q, n). This family of groups parametrizes all quotients of the complex reflection groups G(r, p, n) by scalar subgroups. Our classification is ultimately incomplete, but we provide several necessary and sufficient conditions for generalized involution models to exist in various cases. In the process we solve several intermediate problems concerning the structure of projective reflection groups. We derive a simple criterion for determining whether two groups G(r, p, q, n) and G(r, p′, q′, n) are isomorphic. We also describe explicitly the form of all automorphisms of G(r, p, q, n), outside a finite list of exceptional cases. Building on prior work, this allows us to prove that G(r, p, 1, n) has a generalized involution model if and only if G(r, p, 1, n) ≌ G(r, 1, p, n). We also classify which groups G(r, p, q, n) have generalized involution models when n = 2, or q is odd, or n is odd.

In F. Caselli (Involutory reflection groups and their models, J. Algebra 24:370–393, 2010), a uniform Gelfand model is constructed for all nonexceptional irreducible complex reflection groups which are involutory. Such models can be naturally decomposed into the direct sum of submodules indexed by S n -conjugacy classes, and we present here a general result that relates the irreducible decomposition of these submodules with the projective Robinson–Schensted correspondence. This description also reflects, in a very explicit way, the existence of split representations for these groups.

We study the distribution of the major index with sign on some parabolic quotients of the symmetric group, extending and generalizing simultaneously results Gessel-Simion and Adin-Gessel-Roichman, and on some special trees that we call rakes. We further consider and compute the distribution of the flag-major index on some parabolic quotients of wreath products and other related groups. All these distributions turn out to have very simple factorization formulas.

We introduce the class of projective reflection groups which includes all complex reflection groups. We show that several aspects involving the combinatorics and the representation theory of all non-exceptional irreducible complex reflection groups find a natural description in this wider setting.