Papers

We extend the explicit octonionic approach to groups of type $\mathrm{E}_6$ to the twisted groups ${}^2\mathrm{SE}_{6,K}(F)$ associated with a quadratic Galois extension $K/F$. These groups act on the Albert space over $K$, preserving the Dickson–Freudenthal determinant and a Hermitean form. When the field norm $K^{\times}\to F^{\times}$ is surjective, we give explicit generators and coördinate proofs of the three-orbit theorem on white points and of the stabilisers of representative white vectors. Under this hypothesis we also prove simplicity of the central quotient. Over finite fields we recover the classical orbit lengths and group orders from the stabiliser orders and the total number of white points. Finally, without assuming norm surjectivity, we show that the two isotropic white-point orbits remain unchanged, while the non-isotropic orbits are parametrised by $F^{\times}/\mathrm{N}_{K/F}(K^{\times})$.

The notion of extremal subsets was introduced by Assaf, Dranowski, and González as a natural generalization of Demazure crystals. In this paper, we show that the crystal basis of a finite-dimensional irreducible module over a quantum group of finite type has a unique maximal proper extremal subset, whose complement is the set of elements that generate the whole crystal basis as an extremal subset. We conjecture that this complement consists of all nonzero elements obtained by successively applying the lowering Kashiwara operators to a distinguished element, which is constructed from the lowest element of the crystal basis. We prove this conjecture in type A.

We prove the uniqueness conjecture of Fomin and Neville for their long mutation cycles in all ranks $n\ge4$. More generally, we show that every nucleus-extension quiver has a unique simple cycle in its labeled mutation graph. These quivers allow an acyclic part of arbitrary size outside the nucleus. Our proof uses three mutation statuses determined by a closed nucleus and the full subquivers obtained by deleting one part of its nucleus decomposition at a time from the ambient quiver. We analyze transitions between these statuses to prove that every mutation leaving the prescribed cycle is performed at an exit.

Let $(A,T,B)$ be a splitting tilting triple. We provide a construction of $\tau_2$-tilting $B$-modules from $\tau$-tilting $A$-modules. On the other hand, if $A$ is hereditary, by considering the mutation of a certain two-term silting complex of $A$ and its image under the derived equivalence $\mathbf{R}{\rm Hom}_A(T,-):D^b(A)\to D^b(B)$, we get a three-term silting complex ${\bf W}$ of $B$. Under an explicit non-vanishing condition on the cones of morphisms between the two summands, we prove that ${\rm H}^0\boldsymbol{W}$ is a basic $2$-tilting $B$-module and that it coincides with the module obtained from the first construction. As an application, we generalize the construction of Peng from tilting modules to $\tau$-tilting modules. We also determine the endomorphism algebra of the constructed module as an explicit triangular matrix algebra, which is derived equivalent to $A$ and $B$ whenever the constructed module is $2$-tilting, and give a full semi-orthogonal decomposition of $K^b(\operatorname{proj}A)$.

We consider local cohomology sheaves on compact Hermitian symmetric spaces supported in Schubert varieties, which carry natural structures as mixed Hodge modules. Our main result gives an explicit root-theoretic construction, uniform across Lie types, for the D-simple composition factors and weight filtration on these modules. For Lagrangian and orthogonal Grassmannians, we translate these formulas into combinatorial rules involving strict partitions and shifted Dyck patterns, analogous to existing formulas on ordinary Grassmannians. As an application, we expand upon work of Raicu–Weyman to describe the weight filtration on local cohomology supported in symmetric determinantal and Pfaffian varieties. We also give formulas for the Hodge rational homology level and local cohomological defect of every Schubert variety in a compact Hermitian symmetric space. Finally, we tabulate the local cohomology weight filtrations for all Schubert varieties in the two exceptional Hermitian pairs.

Let $\Lambda=KQ/I$ be a finite-dimensional zero-relation algebra. We encode Crawley–Boevey tree modules over $\Lambda$ by finite rooted trees labelled by arrows of $Q$ and their formal inverses, and construct a deterministic finite bottom-up tree automaton recognizing exactly these encodings. We define an accepted tree to be an automata-induced tree brick when it has no non-trivial factor–image self-overlap, and use Crawley–Boevey's graph-map basis to prove that this is equivalent to brickness of the associated tree module. We also introduce local colourings of $Q_1$ and show that the arrow alphabet can be compressed without changing the tree data, graph maps, or brick property. The optimal number of colours for such a compression is the maximum of the in-degree and out-degree of $Q$. We conclude by asking whether the tree language consisting only of bricks is regular.

We develop a representation-theoretic framework for computing projection constants of finite-dimensional subspaces of $C(K)$, where $K$ is a compact homogeneous $G$-space. Within this framework, we characterize the finite-dimensional $G$-invariant subspaces for which the $G$-equivariant projection is unique. These are precisely the finite orthogonal sums of full isotypic components in the Peter–Weyl decomposition of $L_2(K)$. For such subspaces, averaging shows that this unique $G$-equivariant projection is minimal; it is the restriction to $C(K)$ of the $L_2(K)$-orthogonal projection. Consequently, their projection constants are given by the $L_1$-norm of an explicit reproducing-kernel slice. We apply this method to spaces of $d$-homogeneous polynomials in high dimension. The examples range from Fourier analysis on the torus, through spherical harmonic analysis on real and complex Euclidean spheres, to genuinely noncommutative harmonic analysis on the unitary group, where representation theory becomes indispensable. For all these families, we obtain precise high-dimensional asymptotics at the square-root-of-dimension scale.

The main purpose of this paper is to provide a cohomology theory of restricted Lie triple systems in low degrees and their algebraic interpretations. We define restricted cochains up to degree 5, together with the corresponding coboundary operators. Moreover, we provide interpretations of the low-degree restricted cohomology groups in terms of restricted derivations, extensions of restricted modules, formal deformations, and extensions of restricted Lie triple systems.

Let $G$ be a finite abelian group, and let $k$ be an infinite perfect field containing a primitive root of unity of order the exponent of $G$. We prove that the $G$-equivariant version of Gabber's presentation lemma proposed by Bachmann holds over $k$ if and only if $G$ is cyclic of $p$-power order, for some prime $p$.

The main purpose of this paper is to define and study the notion of restricted Rota-Baxter operator on restricted Lie algebras and restricted Lie triple systems. We provide the relevant properties and the usual connections with pre-Lie structures. We show that the operad of pre-Lie triple systems is the splitting of the operad of Lie triple systems. Moreover, we prove that Jacobson identities hold in pre-Lie triple systems in positive characteristic and then introduce the notion of restricted pre-Lie triple system.

The main purpose of this paper is to introduce and study restricted Lie color triple systems. We consider first restricted Lie color algebras and then provide a definition and relevant properties of restricted Lie color triple systems. Moreover, we introduce and discuss representations of restricted Lie color algebras and restricted Lie color triple systems.

We develop a categorical framework for Lie-Rinehart monoids and their weak modules in a symmetric monoidal category. Using crossed homomorphisms, we construct a natural action of the monoidal category of modules over a Lie monoid on the category of weak Lie-Rinehart modules, thereby obtaining categorical versions of the Shen-Larsson functors. We further characterize the conditions under which the category of weak modules admits a monoidal structure and identify the corresponding condition for the associated functors to be strict monoidal. A dual theory for Lie- Rinehart comonoids and weak comodules is developed using cocrossed homomorphisms. Combining the module and comodule constructions, we obtain a bimodule category structure on the category of weak modules. Finally, we specialize the general framework to the symmetric monoidal category of super vector spaces, recovering Lie-Rinehart superalgebras and their associated Shen-Larsson-type constructions.

In this paper, we construct a Gelfand model for the Okada algebra $O_n(X,Y)$ with generic parameters $X$ and $Y$, on the space of symmetric Okada arc diagrams using a conjugation-type action. The model is constructed inductively by identifying the Okada algebra as a diagram algebra and using the Jones basic construction to obtain a tower of algebras that are themselves Okada algebras at lower levels. We use the model to obtain all the irreducible representations of $O_n(X,Y)$, indexed by the elements of rank $n$ of the Young–Fibonacci lattice, and identify them with the cell modules of $O_n(X,Y)$.

We study the quantum $K$-theory of the flag variety $G/B$. For each minuscule fundamental weight $\varpi$, we construct an explicit relation in the torus-equivariant quantum $K$-theory $QK_T(G/B)$. The relation can be regarded as a quantum deformation of the character of the irreducible representation with highest weight $\varpi$.

We construct commutative rings whose derived categories have non-spatial frames of smashing ideals, thereby answering in the negative questions of Balchin–Stevenson and Aoki.

We consider a double flag variety of the form $L/Q\times G/P$ where $G$ is a connected reductive group and $L$ is a Levi subgroup of $G$. The Levi subgroup $L$ acts diagonally on this double flag variety and a basic problem is to classify pairs of parabolic subgroups $P\subset G$ and $Q\subset L$ such that there are finitely many orbits for the considered action. We solve this problem in the case where $G$ is the general linear group by using representations of quivers.

A qKZ-type additive discrete flat connection in characteristic $p$ has $p$-curvature operators. They are commuting automorphisms of the connection. We present a Bethe-ansatz-type construction of eigensections and eigenvalues of the $p$-curvature operators. An eigensection of $p$-curvature operators is a discrete hypergeometric sum over the finite lattice $\mathbb{Z}^r/p\mathbb{Z}^r$. Thus, the $p$-curvature eigensections are constructed by a finite, discrete analogue of a hypergeometric integral.

We use the series representation of the Bessel functions of type $B_N$ in terms of Jack polynomials and show that $$\lim_{k_1\to\infty} J_{(k_1,k_2)}^B(x,k_1y)^{1/k_1}= 2^N\prod\limits_{i=1}^N \Bigg( e^{\sqrt{1+x_i^2 y_i^2}-1}\cdot \frac{1}{\sqrt{1+x_i^2 y_i^2}+1} \Bigg)$$ for $x,y\in\mathbb R^N$ and $k_2\ge0$. Moreover, the known Laplace-type integral representations for $N\ge1$ and $k_2=0,1/2,1,2$ and for $N=2$ and $k_2>0$ by Rösler and Demni respectively lead to related limits for $$\lim_{k_1 \to \infty} \partial_{x_j} J_{(k_1,k_2)}^B(x,k_1y) /(k_1 \cdot J_{(k_1,k_2)}^B(x,k_1 y)) \quad (j=1,\ldots,N).$$ These limits lead to weak limit results for the associated Bessel processes with drift. For $k_2=1/2,1,2$, these limit results have applications to radial parts of Brownian motions with drift on the $M\times N$-dimensional matrices over $\mathbb R,\mathbb C$, and the quaternions for $M\to\infty$. We also discuss these limits in the Dunkl case $N=1$.

We prove the conjecture of Huang, Jiang, and Oblomkov (HJO) giving a geometric extension of the Rogers–Ramanujan and Andrews–Gordon identities for every torus-knot singularity $X^a=Y^b$ with coprime $1<a<b.$ For a prime power $q$, let $\mathcal{NC}_n^{a,b}(\mathbb F_q)$ denote the set of pairs of commuting nilpotent $n\times n$ matrices $(A,B)$ over $\mathbb F_q$ satisfying $A^a=B^b$. We establish the threefold equality between their normalized counts, the HJO $q$-series $Z_{a,b}$, and the explicit infinite product $P_{a,b}$: \[ \underbrace{\vphantom{\Bigg|} \prod_{m\geq1}(1-q^{-m}) \Biggl(\sum_{n=0}^{\infty} \frac{\lvert\mathcal{NC}_n^{a,b}(\mathbb F_q)\rvert} {\lvert\operatorname{GL}_n(\mathbb F_q)\rvert}\Biggr) }_{\text{point count}} = \underbrace{\vphantom{\Bigg|}Z_{a,b}(q^{-1}) }_{\text{\(q\)-series}} = \underbrace{\vphantom{\Bigg|}P_{a,b}(q^{-1}) }_{\text{infinite product}}. \] Our main result is a stronger finite identity: the rank $N$ HJO sum equals $(q;q)_N$ times the generating function for balanced cylindric partitions with entries bounded by $N$. Taking $N\to\infty$ yields the HJO conjecture. The proof combines the compositional rational shuffle theorem of Bergeron–Garsia–Leven–Xin and Mellit with a multiplicativity theorem for slope operators and a determinantal model for bounded cylindric partitions, linked by a common $q$-difference equation. The finite identity and the HJO conjecture have been formalized unconditionally in Lean by AxiomProver.

Formal Lie groups, which generalize Lie groups in differential geometry, are analogous to formal group schemes in algebraic geometry. In a previous paper, we established the basic theory of formal Lie groups, including a formal Lie theory theorem that identifies formal Lie groups with Lie pairs. In this paper, we develop the foundations of the representation theory of formal Lie groups and Lie pairs. In particular, we construct modules of Lie pairs on function spaces over formal manifolds and prove that the category of representations of a formal Lie group is isomorphic to the category of modules of the corresponding Lie pair.

We classify the automorphism groups of smooth cubic threefolds and fourfolds. We also show that there are $156$ (respectively, $40$) connected families of smooth cubic fourfolds (respectively, threefolds) with specified automorphism group action. For each family, the group action and defining equations are also given. The classification combines both representation theory (based on GAP) and lattice theory (based on SageMath).

We determine the branching of irreducible unitary representations of the basic classical Lie superalgebras $\mathfrak{sl}(m|n)$, $\mathfrak{osp}(m|2n)$, and $F(4)$ under restriction to their even subalgebras. The proof combines Dirac inequalities for the Huang-Pandžić-Dirac operator $\operatorname{D}_{\mathfrak{g},\mathfrak{g}_{\bar{0}}}$ with a super analogue of Freudenthal's recursion derived from the Dirac formalism. We extend the Dirac method to positive systems that are not adapted to the relevant real form by transporting the Dirac inequality through odd reflections and determining the resulting correction terms. This yields finite recursive procedures that determine the occurring even constituents and their branching multiplicities, together with closed formulas for the latter, uniformly in the degree of atypicality. For $\mathfrak{sl}(m|n)$, the corrected Dirac inequalities also give a characterization of unitarity for all positive systems relevant to unitary representations.

We prove a multiplication formula for the Wang-Wei-Zhang cluster character on a Hom-finite $2$-Calabi–Yau Frobenius extriangulated category $\mathcal C$ with a cluster-tilting object under the assumption that its stable category has constructible cones with respect to the induced cluster-tilting object. This formula generalizes those obtained by Wang-Wei-Zhang for $\mathcal C$ in the one-dimensional case and by Keller-Plamondon-Qin for (stably) $2$-Calabi-Yau Frobenius or triangulated categories. As a consequence, we express the frozen term in an Auslander-Reiten mesh with the character of a minimal cosyzygy middle term. For acyclic quivers with principal coefficients, we compute these frozen multiplicities in Higgs categories and obtain explicit principal coefficient mesh relations.

We establish a Jordan decomposition formula for the characteristic functions of the character sheaves on parahoric subgroups defined in \cite{INY25}. For a sufficiently large $q$, these functions are $(-1)^{\dim G}$ times the corresponding deep level Deligne–Lusztig characters, extending \cite{Lu90} to positive depth. We also prove the orthogonality for the generalized deep level Green functions and the character sheaves functions. Moreover, we obtain an explicit expression of characteristic functions of simple character sheaves. As an application, we present a sheaf-theoretic expression to the multiplicity of the $G^{F^2}$ character restricted to $G^{F}$.

We prove that the integral Chow groups $CH^i$ and mixed Hodge degree $H^{2i, (i, i)}$ cohomology groups of really full rank (RFR) sink-recurrent cluster varieties vanish for $i > 0$. In particular this applies to braid varieties and open Richardson varieties in any Lie type. Our main tool is the construction of a stratification of any RFR sink-recurrent cluster variety $\mathcal{A}(\Sigma)$ into (affine spaces times) RFR sink-recurrent cluster varieties of seeds with fewer mutable vertices than $\Sigma$. We employ the theory of Voevodsky motives, and towards this end we prove that the cycle class maps are isomorphisms onto the lowest-weight part of rational Borel-Moore homology for any mixed Tate variety over a number field. We then show that RFR sink-recurrent cluster varieties have mixed Tate and, in fact, split motives. Finally, we use our results to deduce vanishing theorems about the Khovanov-Rozansky homology groups of closures of positive braids and generation properties of the cohomology of closed Richardson, projected Richardson, and brick varieties.

Using ideas of Digne and recent work of Gaitsgory–Rozenblyum–Varshavsky, we show that Lusztig induction commutes with (a minor variant of) the Glauberman correspondence for large prime degree field extensions. This allows one to reduce the Mackey formula in Deligne–Lusztig theory to the case of “large” base fields, in which case the result is due to Bonnafé.

We prove that the $L$-packets for $p$-adic $\mathrm{G}_2$ constructed by Gan and Savin satisfy the endoscopic character identities. In the course of the proof, we also prove twisted endoscopic character identities for functorial lifts from $\mathrm{G}_2$ to $\mathrm{PGSO}_8$ with triality. As a byproduct, we also prove a global multiplicity formula for the discrete automorphic spectrum of $\mathrm{G}_2$ coming from the cuspidal automorphic spectrum of $\mathrm{PGL}_3$.

We prove that the twisted endoscopic character identities between triality $\mathrm{PGSO}_8$ and the endoscopic group $\mathrm{G}_2$ imply the Hiraga-Ichino-Ikeda conjecture on formal degrees for $\mathrm{G}_2$. In the course of proving the main result, we also establish some fundamental results on representation theory of $\mathrm{PGSO}_8$ with triality. A key feature of our approach is an application of the adjoint group of type $E_6$.

We use skein theoretic `recognition' theorems to classify deformations of certain pivotal or ribbon monoidal categories. We first show that a slight generalization of Kuperberg's characterization of quantum $G_2$ shows that any infinitesimal deformation of quantum $G_2$ as a pivotal category arises by varying $q$. This result applies both at generic $q$, and for the category of tilting modules at $q$ a root of unity (provided we exclude a few small roots of unity). Second, we prove a new Kuperberg-like characterization of Deligne's $S_t$ and use it to show that any infinitesimal deformation of $S_t$ (excluding $t=0$) as a ribbon category comes from varying $t$.

Let $\psi$ be a real Arthur parameter and let $\psi_2$ be the unipotent parameter in a fixed Jordan decomposition. Adams, Ionov, Mason-Brown, and Vogan proved that the microlocal packet of $\psi$ is contained in the support of the two-step Jordan induction of the packet of $\psi_2$, and conjectured equality. We prove the reverse inclusion. The argument first passes to a sufficiently positive translate, where the relevant connected components of the ABV spaces have a common flag-variety model. Perverse devissage reduces the problem to a singular-support implication, which follows from the incidence calculation underlying the AIMV microstalk comparison. Coherent continuation for individual packet members then returns the result to the original parameter.