Inventiones mathematicae最新文献

The totally nonnegative flag variety was introduced by Lusztig. It has enriched combinatorial, geometric, and Lie-theoretic structures. In this paper, we introduce a (new) (J)-total positivity on the full flag variety of an arbitrary Kac-Moody group, generalizing the (ordinary) total positivity.

We show that the (J)-totally nonnegative flag variety has a cellular decomposition into totally positive (J)-Richardson varieties. Moreover, each totally positive (J)-Richardson variety admits a favorable decomposition, called a product structure. Combined with the generalized Poincare conjecture, we prove that the closure of each totally positive (J)-Richardson variety is a regular CW complex homeomorphic to a closed ball. Moreover, the (J)-total positivity on the full flag provides a model for the (ordinary) totally nonnegative partial flag variety. As a consequence, we prove that the closure of each (ordinary) totally positive Richardson variety is a regular CW complex homeomorphic to a closed ball, confirming conjectures of Galashin, Karp and Lam in (Adv. Math. 351:614–620, 2019). We also show that the link of the totally nonnegative part of (U^{-}) for any Kac-Moody group forms a regular CW complex. This generalizes the result of Hersh (Invent. Math. 197(1):57–114, 2014) for reductive groups.

We develop new techniques to compute the weighted (L^{2})-cohomology of quasi-fibered boundary metrics (QFB-metrics). Combined with the decay of (L^{2})-harmonic forms obtained in a companion paper, this allows us to compute the reduced (L^{2})-cohomology for various classes of QFB-metrics. Our results applies in particular to the Nakajima metric on the Hilbert scheme of (n) points on (mathbb{C}^{2}), for which we can show that the Vafa-Witten conjecture holds. Using the compactification of the monopole moduli space announced by Fritzsch, the first author and Singer, we can also give a proof of the Sen conjecture for the monopole moduli space of magnetic charge 3.

We prove that if (u:mathbb{R}^{n}to mathbb{R}) is strongly convex, then for every (varepsilon >0) there is a strongly convex function (vin C^{2}(mathbb{R}^{n})) such that (|{uneq v}|<varepsilon ) and (Vert u-vVert _{infty}<varepsilon ).

This note corrects an error in our paper Anosov flows, growth rates on covers and group extensions of subshifts, Invent. Math. 223:445–483, 2021. This leaves our main results, Theorem 1.1, Corollary 1.2, Theorem 1.3 and Theorem 5.1, unchanged. We also fill a gap in the arguments presented in Sect. 9; this requires a small modification to the results in this section.

The Benjamin–Ono equation is shown to be well-posed, both on the line and on the circle, in the Sobolev spaces (H^{s}) for (s>-tfrac{1}{2}). The proof rests on a new gauge transformation and benefits from our introduction of a modified Lax pair representation of the full hierarchy. As we will show, these developments yield important additional dividends beyond well-posedness, including (i) the unification of the diverse approaches to polynomial conservation laws; (ii) a generalization of Gérard’s explicit formula to the full hierarchy; and (iii) new virial-type identities covering all equations in the hierarchy.

Let (mathfrak {g}) be a symmetrisable Kac–Moody algebra and (V) an integrable (mathfrak {g})–module in category (mathcal {O}). We show that the monodromy of the (normally ordered) rational Casimir connection on (V) can be made equivariant with respect to the Weyl group (W) of (mathfrak {g}), and therefore defines an action of the braid group (mathcal {B}_{W}) on (V). We then prove that this action is canonically equivalent to the quantum Weyl group action of (mathcal {B}_{W}) on a quantum deformation of (V), that is an integrable, category (mathcal {O}) module (mathcal {V}) over the quantum group (U_{hbar }mathfrak {g}) such that (mathcal {V}/hbar mathcal {V}) is isomorphic to (V). This extends a result of the second author which is valid for (mathfrak {g}) semisimple.

We introduce a new family of real simple modules over the quantum affine algebras, called the affine determinantial modules, which contains the Kirillov-Reshetikhin (KR)-modules as a special subfamily, and then prove T-systems among them which generalize the T-systems among KR-modules and unipotent quantum minors in the quantum unipotent coordinate algebras simultaneously. We develop new combinatorial tools: admissible chains of (i)-boxes which produce commuting families of affine determinantial modules, and box moves which describe the T-system in a combinatorial way. Using these results, we prove that various module categories over the quantum affine algebras provide monoidal categorifications of cluster algebras. As special cases, Hernandez-Leclerc categories (mathscr{C}_{{mathfrak{g}}}^{0}) and (mathscr{C}_{{mathfrak{g}}}^{-}) provide monoidal categorifications of the cluster algebras for an arbitrary quantum affine algebra.

Catalan functions, the graded Euler characteristics of certain vector bundles on the flag variety, are a rich class of symmetric functions which include (k)-Schur functions and parabolic Hall-Littlewood polynomials. We prove that Catalan functions indexed by partition weight are the characters of (U_{q}(widehat{mathfrak{sl}}_{ell }))-generalized Demazure crystals as studied by Lakshmibai-Littelmann-Magyar and Naoi. We obtain Schur positive formulas for these functions, settling conjectures of Chen-Haiman and Shimozono-Weyman. Our approach more generally gives key positive formulas for graded Euler characteristics of certain vector bundles on Schubert varieties by matching them to characters of generalized Demazure crystals.

We prove a sharp quantitative version of the Faber–Krahn inequality for the short-time Fourier transform (STFT). To do so, we consider a deficit (delta (f;Omega )) which measures by how much the STFT of a function (fin L^{2}(mathbb{R})) fails to be optimally concentrated on an arbitrary set (Omega subset mathbb{R}^{2}) of positive, finite measure. We then show that an optimal power of the deficit (delta (f;Omega )) controls both the (L^{2})-distance of (f) to an appropriate class of Gaussians and the distance of (Omega ) to a ball, through the Fraenkel asymmetry of (Omega ). Our proof is completely quantitative and hence all constants are explicit. We also establish suitable generalizations of this result in the higher-dimensional context.

Previously, the authors proved that the presentation complex of a one-relator group (G) satisfies a geometric condition called negative immersions if every two-generator, one-relator subgroup of (G) is free. Here, we prove that one-relator groups with negative immersions are coherent, answering a question of Baumslag in this case. Other strong constraints on the finitely generated subgroups also follow such as, for example, the co-Hopf property. The main new theorem strengthens negative immersions to uniform negative immersions, using a rationality theorem proved with linear-programming techniques.