Search

Let S be the polynomial ring on the space of non-square generic matrices or the space of odd-sized skew-symmetric matrices, and let I be the determinantal ideal of maximal minors or Pf the ideal of sub-maximal Pfaffians, respectively. Using desingularizations and representation theory of the general linear group we expand upon work of Raicu–Weyman–Witt [Adv. Math. 250 (2014), pp. 596–610] to determine the S-module structures of Ext^j_S(S/I^t, S) and Ext^j_S(S/Pf^t, S), from which we get the degrees of generators of these Ext modules. As a consequence, via graded local duality we answer a question of Wenliang Zhang [J. Pure Appl. Algebra 225 (2021), Paper No. 106789] on the socle degrees of local cohomology modules of the form H^j_\mathfrak {m}(S/I^t). [ABSTRACT FROM AUTHOR]

In this paper the authors introduce an analogue of the nilpotent cone, N, for a classical Lie superalgebra, g, that generalizes the definition for the nilpotent cone for semisimple Lie algebras. For a classical simple Lie superalgebra, g = g0 ⊕ g1 with Lie G0 = g0, it is shown that there are finitely many G0-orbits on N. Later the authors prove that the Duflo-Serganova commuting variety, X, is contained in N for any classical simple Lie superalgebra. Consequently, our finiteness result generalizes and extends the work of Duflo-Serganova on the commuting variety. Further applications are given at the end of the paper. [ABSTRACT FROM AUTHOR]

It is well known that every homology functor on the stable homotopy category is representable, so of the form E*t(X) = Π*(E Λ X) for some spectrum E. However, Christensen, Keller, and Neeman (2001) have exhibited simple triangulated categories, such as the derived category of k[x, y] for sufficiently large fields k, for which not every homology functor is representable. In this paper, we show that this failure of Brown representability does not happen on the model category level. That is, we show that a homology theory is representable if and only if it lifts to a well-behaved functor on the model category level. We also show that, for a reasonable model category M, every functor that has the same formal properties as a functor of the form X → X ⊗ E for some cofibrant E is naturally weakly equivalent to a functor of that form. This is closely related to the Eilenberg-Watts theorem in algebra, which proves that every functor with the same formal properties as the tensor product with a fixed object is isomorphic to such a functor. [ABSTRACT FROM AUTHOR]

The smooth hermitian representations of a split reductive p-adic group whose restriction to a maximal hyperspecial compact subgroup contain a single K-type with Iwahori fixed vectors have been studied in a paper by Barbasch and Moy (1999) in the more general setting of modules for graded affine Hecke algebras with parameters. We show that every such one K-type module has nonzero Dirac cohomology (in the sense of a paper by Barbasch, Ciubotaru and Trapa), and use Dirac operator techniques to determine the semisimple part of the Langlands parameter for these modules, thus completing their classification. [ABSTRACT FROM AUTHOR]

In this paper, we present a direct bijective proof of the hook-length formula for standard immaculate tableaux, which arose in the study of noncommutative symmetric functions. Our proof is along the spirit of Novelli, Pak and Stoyanovskiĭ's combinatorial proof of the hook-length formula for standard Young tableaux. [ABSTRACT FROM AUTHOR]

In this paper, we prove the K- and L-theoretical Isomorphism Conjecture for Baumslag-Solitar groups with coefficients in an additive category. [ABSTRACT FROM AUTHOR]

Let F be a non-Archimedean local field of characteristic 0 and G = SL(n,F). Let (π,W) be an irreducible smooth self-dual representation G. The space W of π carries a non-degenerate G-invariant bilinear form (,) which is unique up to scaling. The form (,) is easily seen to be symmetric or skew-symmetric and we set ε(π) = ±1 accordingly. In this article, we show that ε(π) = 1 when π is an Iwahori spherical representation of G. [ABSTRACT FROM AUTHOR]

Let M be a cusped hyperbolic 3-manifold, e.g. a knot complement. Thurston showed that the space of deformations of its fundamental group in PGL(2, C) (up to conjugation) is of complex dimension the number v of cusps near the hyperbolic representation. It seems natural to ask whether some representations remain discrete after deformation. The answer is generically not. A simple reason for it lies inside the cusps: the degeneracy of the peripheral representation (i.e. representations of fundamental groups of the v peripheral tori). They indeed generically become non-discrete, except for a countable set. This last set corresponds to hyperbolic Dehn surgeries on M, for which the peripheral representation is no more faithful. We work here in the framework of PGL(ra, C). The hyperbolic structure lifts, via the ra-dimensional irreducible representation, to a representation Pgeom. We know from the work of Menal-Ferrer and Porti that the space of deformations of Pgeom has complex dimension (ra -- 1)v . We prove here that, unlike the PGL(2)-case, the generic behaviour becomes the discreteness (and faithfulness) of the peripheral representation: in a neighbourhood of the geometric representation, the non-discrete peripheral representations are contained in a real analytic subvariety of codimension ≥ 1. [ABSTRACT FROM AUTHOR]

Star-fundamental algebras are special finite dimensional algebras with involution * over an algebraically closed field of characteristic zero defined in terms of multialternating *-polynomials. We prove that the upper-block matrix algebras with involution introduced in Di Vincenzo and La Scala [J. Algebra 317 (2007), pp. 642–657] are star-fundamental. Moreover, any finite dimensional algebra with involution contains a subalgebra mapping homomorphically onto one of such algebras. We also give a characterization of star-fundamental algebras through the representation theory of the symmetric group. [ABSTRACT FROM AUTHOR]

We determine the homological residue fields, in the sense of tensor-triangular geometry, in a series of concrete examples ranging from topological stable homotopy theory to modular representation theory of finite groups. [ABSTRACT FROM AUTHOR]

Let G be a compact connected semisimple Lie group, let K be a closed subgroup of G, let Γ be a finite subgroup of G, and let π be a finite-dimensional representation of K. For π in the unitary dual Ĝ of G, denote by nΓ (π) its multiplicity in L2(Γ\G). We prove a strong multiplicity one theorem in the spirit of Bhagwat and Rajan, for the nΓ (π) for π in the set Ĝπ of irreducible π-spherical representations of G. More precisely, for Γ and Γ' finite subgroups of G, we prove that if nΓ(π) = nΓ'(π) for all but finitely many π ∈ Ĝπ, then Γ and Γ' are π-representation equivalent, that is, nΓ(π) = nΓ'(π) for all π ∈ Ĝπ. Moreover, when Ĝ_π can be written as a finite union of strings of representations, we prove a finite version of the above result. For any finite subset \widehat {Fπ of Ĝπ verifying some mild conditions, the values of the nΓ (π) for π ∈ Fπ determine the nΓ (π)'s for all π ∈ Ĝπ. In particular, for two finite subgroups Γ and Γ' of G, if nΓ (π) = nΓ'(π) for all π ∈ Fπ, then the equality holds for every π ∈ Ĝπ. We use algebraic methods involving generating functions and some facts from the representation theory of G. [ABSTRACT FROM AUTHOR]

Let V be a rational, selfdual, C2-cofinite vertex operator algebra of CFT type, and G a finite automorphism group of V. It is proved that the kernel of the representation of the modular group on twisted conformal blocks associated to V and G is a congruence subgroup. In particular, the q-character of each irreducible twisted module is a modular function on the same congruence subgroup. In the case V is the Frenkel-Lepowsky-Meurman's moonshine vertex operator algebra and G is the monster simple group, the generalized McKay-Thompson series associated to any commuting pair in the monster group is a modular function. [ABSTRACT FROM AUTHOR]

In this note we consider the complex representation theory of FId, a natural generalization of the category FI of finite sets and injections. We prove that finitely generated FId-modules exhibit behaviors in the spirit of Church-Farb representation stability theory, generalizing a theorem of Church, Ellenberg, and Farb which connects finite generation of FI-modules to representation stability. [ABSTRACT FROM AUTHOR]

We classify the modular representations of a cyclic group of prime order whose corresponding rings of coinvariants are Poincaré duality algebras. It turns out that these algebras are actually complete intersections. For other representations we demonstrate that the dimension of the top degree of the coinvariants grows at least linearly with respect to the number of summands of dimension at least four in the representation. [ABSTRACT FROM AUTHOR]

Langlands' beyond endoscopy proposal for establishing functoriality motivates interesting and concrete problems in the representation theory of algebraic groups. We study these problems in a setting related to the Langlands L-functions L(s, π, ⊗³), where π is a cuspidal automorphic representation of GLn(AF) and F is a global field. [ABSTRACT FROM AUTHOR]

Given a pair of n-dimensional complex Galois representations over Q, we define their matching density to be the density, if it exists, of the set of places at which the traces of Frobenius of the two Galois representations are equal. We will show that the set of matching densities of such pairs of irreducible Galois representations (for all n) is dense in the interval [0, 1]. We then discuss the automorphic analogue of this problem. [ABSTRACT FROM AUTHOR]

In this note we prove a certain multiplicity formula regarding the restriction of an irreducible admissible genuine representation of a 2-fold cover GL2 (F) of GL2 (F) to the 2-fold cover SL2 (F) of SL2 (F), and find in particular that this multiplicity may not be one, a result that was recently observed for certain principal series representations in the work of Szpruch (2013). The proofs follow the standard path via Waldspurger's analysis of theta correspondence between SL2(F) and PGL2(F). [ABSTRACT FROM AUTHOR]

We present a characterization of the sets that appear as Fourier spectra of measures in terms of the existence of a strongly continuous representation of the ambient group that has a wandering vector for the given set. [ABSTRACT FROM AUTHOR]

An Archimedean l-group is epicomplete if it is divisible and σ complete, both laterally and conditionally. Under various circumstances it has been shown that epicompleteness implies the existence of a compatible reduced f-ring multiplication; the question has arisen whether or not this is always true. We show that a set-theoretic condition weaker than the continuum hypothesis implies "not", and conjecture the converse. The examples also fail decent representation and existence of some other compatible operations. [ABSTRACT FROM AUTHOR]

We construct indecomposable modules for the 0-Hecke algebra whose characteristics are the dual immaculate basis of the quasi-symmetric functions. [ABSTRACT FROM AUTHOR]

For every integer µ ≥ 3, there exists a function fµ: N+ → N+ such that the following holds: (1) fµ(k) = 2k -- µ for k large enough; (2) if ... is a finite nonempty collection of subalgebras of P(X) such that ∩... B is not fµ (#(...))-saturated, for all nonempty ... ⊆ ..., then ∪ ≄ P(X). [ABSTRACT FROM AUTHOR]

We introduce a reproducing kernel structure for Hilbert spaces of functions where differences of point evaluations are bounded. The associated reproducing kernels are characterized in terms of conditionally negative functions. [ABSTRACT FROM AUTHOR]