Selecta Mathematica最新文献

We employ general parabolic recursion methods to demonstrate the recently devised hypercube formula for Kazhdan-Lusztig polynomials of (S_n), and establish its generalization to the full setting of a finite Coxeter system through algebraic proof. We introduce procedures for positive decompositions of q-derived Kazhdan–Lusztig polynomials within this setting, that utilize classical Hecke algebra positivity phenomena of Dyer-Lehrer and Grojnowski–Haiman. This leads to a distinct algorithmic approach to the subject, based on induction from a parabolic subgroup. We propose suitable weak variants of the combinatorial invariance conjecture and verify their validity for permutation groups.

We establish identities for the composition (T_{k,n}(|widehat{gdsigma }|^2)), where (gmapsto widehat{gdsigma }) is the Fourier extension operator associated with a general smooth k-dimensional submanifold of ({mathbb {R}}^n), and (T_{k,n}) is the k-plane transform. Several connections to problems in Fourier restriction theory are presented.

We prove the Morrison–Kawamata cone conjecture for projective primitive symplectic varieties with ({mathbb Q})-factorial and terminal singularities with (b_2ge 5), from which we derive for instance the finiteness of minimal models of such varieties, up to isomorphisms. To prove the conjecture we establish along the way some results on the monodromy group which may be interesting in their own right, such as the fact that reflections in prime exceptional divisors are integral Hodge monodromy operators which, together with monodromy operators provided by birational transformations, yield a semidirect product decomposition of the monodromy group of Hodge isometries.

We give a recursive method for computing all values of a basis of Whittaker functions for unramified principal series invariant under an Iwahori or parahoric subgroup of a split reductive group G over a nonarchimedean local field F. Structures in the proof have surprising analogies to features of certain solvable lattice models. In the case (G=textrm{GL}_r) we show that there exist solvable lattice models whose partition functions give precisely all of these values. Here ‘solvable’ means that the models have a family of Yang–Baxter equations which imply, among other things, that their partition functions satisfy the same recursions as those for Iwahori or parahoric Whittaker functions. The R-matrices for these Yang–Baxter equations come from a Drinfeld twist of the quantum group (U_q(widehat{mathfrak {gl}}(r|1))), which we then connect to the standard intertwining operators on the unramified principal series. We use our results to connect Iwahori and parahoric Whittaker functions to variations of Macdonald polynomials.

Consider a finite group G acting on a graded Noetherian k-algebra S, for some field k of characteristic p; for example S might be a polynomial ring. Regard S as a kG-module and consider the multiplicity of a particular indecomposable module as a summand in each degree. We show how this can be described in terms of homological algebra and how it is linked to the geometry of the group action on the spectrum of S.

This paper is the third in a series of four papers aiming to describe the (almost integral) Chow ring of (overline{mathcal {M}}_3), the moduli stack of stable curves of genus 3. In this paper, we compute the Chow ring of (widetilde{{mathcal {M}}}_3^7) with ({mathbb {Z}}[1/6])-coefficients.

We determine the rational class and Picard groups of the moduli space of stable logarithmic maps in genus zero, with target projective space relative a hyperplane. For the class group we exhibit an explicit basis consisting of boundary divisors. For the Picard group we exhibit a spanning set indexed by piecewise-linear functions on the tropicalisation. In both cases a complete set of boundary relations is obtained by pulling back the WDVV relations from the space of stable curves. Our proofs hinge on a controlled technique for manufacturing test curves in logarithmic mapping spaces, opening up the topology of these spaces to further study.

We introduce a new method for precisely relating algebraic structures in a presheaf category and judgements of its internal type theory. The method provides a systematic way to organise complex diagrammatic reasoning and generalises the well-known Kripke-Joyal forcing for logic. As an application, we prove several properties of algebraic weak factorisation systems considered in Homotopy Type Theory.

The affine Grassmannian associated to a reductive group ({textbf{G}}) is an affine analogue of the usual flag varieties. It is a rich source of Poisson varieties and their symplectic resolutions. These spaces are examples of conical symplectic resolutions dual to the Nakajima quiver varieties. We study the cohomological stable envelopes of Maulik and Okounkov (Astérisque 408:ix+209, 2019) in this family. We construct an explicit recursive relation for the stable envelopes in the ({textbf{G}}= textbf{PSL}_{2}) case and compute the first-order correction in the general case. This allows us to write an exact formula for multiplication by a divisor.

Let (G, H) be one of the equal rank reductive dual pairs (left( Mp_{2n},O_{2n+1} right) ) or (left( U_n,U_n right) ) over a nonarchimedean local field of characteristic zero. It is well-known that the theta correspondence establishes a bijection between certain subsets, say (widehat{G}_theta ) and (widehat{H}_theta ), of the tempered duals of G and H. We prove that this bijection arises from an equivalence between the categories of representations of two (C^*)-algebras whose spectra are (widehat{G}_theta ) and (widehat{H}_theta ). This equivalence is implemented by the induction functor associated to a Morita equivalence bimodule (in the sense of Rieffel) which we construct using the oscillator representation. As an immediate corollary, we deduce that the bijection is functorial and continuous with respect to weak inclusion. We derive further consequences regarding the transfer of characters and preservation of formal degrees.