Let G be a connected simply-connected simple algebraic group over C and let T be a maximal torus, B superset of T a Borel subgroup and K a maximal compact subgroup. Then, the product in the (algebraic) based loop group Omega(K) gives rise to a comultiplication in the topological T-equivariant K-ring Ktop T (Omega(K)). Recall that Omega(K) is identified with the affine Grassmannian X (of G) and hence we get a comultiplication in Ktop T (X). Dualizing, one gets the Pontryagin product in the T-equivariant Khomology K0T (X), which in-turn gets identified with the convolution product (due to S. Kato). Now, Ktop T(X) has a basis {xi w} over the representation ring R(T) given by the ideal sheaves corresponding to the finite codimension Schubert varieties Xw in X. We make a positivity conjecture on the comultiplication structure constants in the above basis. Using some results of Kato, this conjecture gives rise to an equivalent conjecture on the positivity of the multiplicative structure constants in T-equivariant quantum K-theory QKT(G/B) in the Schubert basis. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

We study non-existence of non-constant regular maps from a partial flag variety $X=G/P$ to another partial flag variety $X'=G'/P'$ and prove that there does not exist any non-constant regular map from any partial non-complete flag variety $X$ to any complete flag variety $X'=G'/B'$. We also formulate a general conjecture on the non-existence of non-constant regular maps from $X\to X'$.

In this paper, we are interested in the decomposition of the tensor product of two representations of a symmetrizable Kac-Moody Lie algebra g, and more precisely in the tensor cone of g. Let P + be the set of dominant integral weights. For $\lambda$ $\in$ P + , L($\lambda$) denotes the (irreducible) integrable, highest weight representation of g with highest weight $\lambda$. Let P +,Q be the rational convex cone generated by P +. Consider the tensor cone $\Gamma$(g) := {($\lambda$ 1 , $\lambda$ 2 , $\mu$) $\in$ P 3 +,Q : $\exists$N $\ge$ 1 such that L(N$\mu$) $\subset$ L(N$\lambda$ 1)$\otimes$L(N$\lambda$ 2)}. If g is finite dimensional, $\Gamma$(g) is a polyhedral convex cone described in [BK06] by an explicit finite list of inequalities. In [Res10] this list of inequalities is proved to be irredundant: each inequality corresponds to a codimension one face. In general, $\Gamma$(g) is neither polyhedral, nor closed. Brown-Kumar [BK14] obtained a list of inequalities that describe $\Gamma$(g) conjecturally. Here, we prove that each of Brown-Kumar's inequalities corresponds to a codimension one face of $\Gamma$(g).

We prove sign-alternation of the product structure constants in the basis dual to the basis consisting of the structure sheaves of Schubert varieties in the torus-equivariant Grothendieck group of coherent sheaves on the partial flag varieties G/P associated to an arbitrary symmetrizable Kac-Moody group G, where P is any parabolic subgroup of finite type. This extends the previous work of Kumar from G/B to G/P. When G is of finite type, i.e., it is a semisimple group, then it was proved by Anderson-Griffeth-Miller.

The main result of this note asserts that a strong form of the Matroid Minor Conjecture due to Draisma is not true, that is, there exist properly ascending chains of $S_\infty $-stable ideals in the affine coordinate ring of the affine infinite Grassmannian, where $S_\infty $ is the infinite symmetric group. In fact, we explicitly construct such an ascending chain. His conjectures on topological Noetherian property for the affine infinite Grassmannian remain open though.

Let C be a connected reductive group over the complex numbers and let T subset of C be a maximal torus. For any t is an element of T of finite order and any irreducible representation V (lambda) of C of highest weight lambda, we determine the character ch(t, V (lambda)) by using the Lefschetz Trace Formula due to Atiyah-Singer and explicitly determining the connected components and their normal bundles of the fixed point subvariety (C/P)t subset of C/P (for any parabolic subgroup P). This together with Wirtinger's theorem gives an asymptotic formula for ch(t, V (n lambda)) when n goes to infinity. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar

Lusztig defined an abelian category C-k of a class of representations of a multi-loop algebra and asked various questions connecting it to the modular representation theory of simple algebraic groups in char. p > 0. The aim of this paper is to show that some of these questions have negative answer.

Let Γ be a finite group acting on a simple Lie algebra 𝔤 and acting on a s-pointed projective curve (Σ , p⃗={p_1, … , p_s}) faithfully (for s≥ 1 ). Also, let an integrable highest weight module ℋ_c(λ _i) of an appropriate twisted affine Lie algebra determined by the ramification at p_i with a fixed central charge c is attached to each p_i . We prove that the space of twisted conformal blocks attached to this data is isomorphic to the space associated to a quotient group of Γ acting on 𝔤 by diagram automorphisms and acting on a quotient of Σ . Under some mild conditions on ramification types, we prove that calculating the dimension of twisted conformal blocks can be reduced to the situation when Γ acts on 𝔤 by diagram automorphisms and covers of ℙ^1 with 3 marked points. Assuming a twisted analogue of Teleman’s vanishing theorem of Lie algebra homology, we derive an analogue of the Kac–Walton formula and the Verlinde formula for general Γ -curves (with mild restrictions on ramification types). In particular, if the Lie algebra 𝔤 is not of type D_4 , there are no restrictions on ramification types.

Following some work of Aluffi–Mihalcea–Schürmann–Su for the CSM classes of Schubert cells and some elaborate computer calculations by Rimanyi and Mihalcea, I conjecture that the CSM classes of the Richardson cells expressed in the Schubert basis have nonnegative coefficients. This conjecture was principally motivated by a new product $\square $ coming from the Segre-SM classes in the cohomology of flag varieties (such that the associated Gr of this product is the standard cup product) and the conjecture that the structure constants of this new product $\square $ in the standard Schubert basis have alternating sign behavior. I prove that this conjecture on the sign of the structure constants of $\square $ would follow from my above positivity conjecture about the CSM classes of Richardson cells.

The explicit Verlinde formula for the dimension of conformal blocks, attached to a marked projective curve $\Sigma$, a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$ and integrable highest weight modules of a fixed central charge of the corresponding affine Lie algebra $\hat{L}(\mathfrak{g})$ attached to the marked points, requires (among several other important ingredients) a Lie algebra cohomology vanishing result due to C. Teleman for the positive part $\hat{L}^+(\mathfrak{g})$ with coefficients in the tensor product of an integrable highest weight module with copies of finite dimensional evaluation modules. The aim of this paper is to extend this result of Teleman to a twisted setting where $\mathfrak{g}$ is endowed with a special automorphism $\sigma$ and the curve $\Sigma$ is endowed with the action of $\sigma$. In this general setting, the affine Lie algebra gets replaced by twisted affine Lie algebras. The crucial ingredient (as in Teleman) is to prove a certain Nakano Identity.

Kostant asked the following question: Let 𝔤 be a simple Lie algebra over the complex numbers. Let λ be a dominant integral weight. Then, V(λ ) is a component of V(ρ )⊗ V(ρ ) if and only if λ≤ 2 ρ under the usual Bruhat–Chevalley order on the set of weights. In an earlier work with R. Chirivi and A. Maffei, the second author gave an affirmative answer to this question up to a saturation factor. The aim of the current work is to extend this result to untwisted affine Kac–Moody Lie algebra 𝔤 associated with any simple Lie algebra 𝔤 (up to a saturation factor). In fact, we prove the result for affine sl_n without any saturation factor. Our proof requires some additional techniques including the Goddard–Kent–Olive construction and study of the characteristic cone of non-compact polyhedra.

We present VeriAbsL, a reachability verifier that performs verification in three stages. First, it slices the input code using a combination of two slicers, then it verifies the slices using predicted strategies, and at last, it composes the result of verifying the individual slices. We introduce a novel shallow slicing technique that uses variable reference information of the program, and data and control dependencies of the entry function to generate slices. We also introduce a novel strategy prediction technique that uses machine learning to predict a strategy. It uses boolean features to describe a program to a neural network that predicts a strategy. We use the portfolio of VeriAbs, a reachabiltiy verifier with manually defined strategies. In sv-comp 2023, VeriAbsL verified 227 (Without witness validation.) more programs than VeriAbs, and 475 (Without witness validation.) programs that VeriAbs could not verify.

We study the spaces of twisted conformal blocks attached to a $\Gamma$ -curve $\Sigma$ with marked $\Gamma$ -orbits and an action of $\Gamma$ on a simple Lie algebra $\mathfrak {g}$ , where $\Gamma$ is a finite group. We prove that if $\Gamma$ stabilizes a Borel subalgebra of $\mathfrak {g}$ , then the propagation theorem and factorization theorem hold. We endow a flat projective connection on the sheaf of twisted conformal blocks attached to a smooth family of pointed $\Gamma$ -curves; in particular, it is locally free. We also prove that the sheaf of twisted conformal blocks on the stable compactification of Hurwitz stack is locally free. Let $\mathscr {G}$ be the parahoric Bruhat–Tits group scheme on the quotient curve $\Sigma /\Gamma$ obtained via the $\Gamma$ -invariance of Weil restriction associated to $\Sigma$ and the simply connected simple algebraic group $G$ with Lie algebra $\mathfrak {g}$ . We prove that the space of twisted conformal blocks can be identified with the space of generalized theta functions on the moduli stack of quasi-parabolic $\mathscr {G}$ -torsors on $\Sigma /\Gamma$ when the level $c$ is divisible by $|\Gamma |$ (establishing a conjecture due to Pappas and Rapoport).

We determine the Lie subalgebra gnil of a Borcherds symmetrizable generalized Kac-Moody Lie algebra g generated by ad-locally nilpotent elements and show that it is ‘essentially’ the same as the Levi subalgebra of g with its simple roots precisely the real simple roots of g.

Let g \mathfrak {g} be an affine Kac-Moody Lie algebra and let λ , μ \lambda , \mu be two dominant integral weights for g \mathfrak {g} . We prove that under some mild restriction, for any positive root β \beta , V ( λ ) ⊗ V ( μ ) V(\lambda )\otimes V(\mu ) contains V ( λ + μ − β ) V(\lambda +\mu -\beta ) as a component, where V ( λ ) V(\lambda ) denotes the integrable highest weight (irreducible) g \mathfrak {g} -module with highest weight λ \lambda . This extends the corresponding result by Kumar from the case of finite dimensional semisimple Lie algebras to the affine Kac-Moody Lie algebras. One crucial ingredient in the proof is the action of Virasoro algebra via the Goddard-Kent-Olive construction on the tensor product V ( λ ) ⊗ V ( μ ) V(\lambda )\otimes V(\mu ) . Then, we prove the corresponding geometric results including the higher cohomology vanishing on the G \mathcal {G} -Schubert varieties in the product partial flag variety G / P × G / P \mathcal {G}/\mathcal {P}\times \mathcal {G}/\mathcal {P} with coefficients in certain sheaves coming from the ideal sheaves of G \mathcal {G} -sub-Schubert varieties. This allows us to prove the surjectivity of the Gaussian map.

As mentioned in Chapter 3, to determine the dimension of the space of vacua on a genus-g curve, it suffices to determine it on the projective line with three marked points. To achieve this, a general algebraic framework in the form of a fusion ring of the simple Lie algebra g at level c is introduced in this chapter. It is a finite rank-reduced algebra. We determine its set of characters explicitly by using the combinatorics of the affine Weyl group and the affine analogue of the Borel--Weil--Bott theorem, as well as a Lie algebra cohomology vanishing result of Teleman. Once we have explicitly determined the characters of the fusion ring (as we have), one of the most important results of the book -- the Verlinde dimension formula -- follows easily by using simple representation theory for finite groups.

In 1988, E. Verlinde gave a remarkable conjectural formula for the dimension of conformal blocks over a smooth curve in terms of representations of affine Lie algebras. Verlinde's formula arose from physical considerations, but it attracted further attention from mathematicians when it was realized that the space of conformal blocks admits an interpretation as the space of generalized theta functions. A proof followed through the work of many mathematicians in the 1990s. This book gives an authoritative treatment of all aspects of this theory. It presents a complete proof of the Verlinde formula and full details of the connection with generalized theta functions, including the construction of the relevant moduli spaces and stacks of G-bundles. Featuring numerous exercises of varying difficulty, guides to the wider literature and short appendices on essential concepts, it will be of interest to senior graduate students and researchers in geometry, representation theory and theoretical physics.

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We define the stack of G-bundles over smooth projective curves for a reductive algebraic group G as well as the stack of quasi-parabolic G-bundles over s-pointed smooth curves. Both of these stacks are smooth algebraic stacks. We define a tautological family of G-bundles over any smooth curve ? parameterized by the infinite Grassmannian. Then, we prove the Uniformization Theorem for both of the above two stacks. An important ingredient in the proof of the above two uniformization theorems is a result due to Drinfeld--Simpson asserting that for a family of G-bundles over ? parameterized by a scheme S, the pull-back of the family to some étale cover is trivial restricted to any affine open subset of ?. In particular, the uniformization theorems give a bijective parameterization of G-bundles as well as quasi-parabolic G-bundles over ?.

We introduce new notions in elliptic Schubert calculus: the (twisted) Borisov-Libgober classes of Schubert varieties in general homogeneous spaces G/P. While these classes do not depend on any choice, they depend on a set of new variables. For the definition of our classes we calculate multiplicities of some divisors in Schubert varieties, which were only known for full flag varieties before. Our approach leads to a simple recursions for the elliptic classes. Comparing this recursion with R-matrix recursions of the so-called elliptic weight functions of Rimanyi-Tarasov-Varchenko we prove that weight functions represent elliptic classes of Schubert varieties.