Papers

In this paper, first we give a new characterization of the cohomology of pre-Lie algebras using the Chevalley-Eilenberg cohomology associated to a morphism from the operad of Lie algebras to Hadamard product of the operad of pre-Lie algebras and its Koszul dual operad. Then we apply the same approach to study the cohomology of Novikov algebras, and give the cochain complex explicitly. The cochain complex of the underlying pre-Lie algebra is shown to be isomorphic to the quotient of the cochain complex of a Novikov algebra. Consequently, there is a long exact sequence connecting the cohomologies of a Novikov algebra and the underlying pre-Lie algebra. The cohomology of a Novikov algebra with coefficients in a representation is introduced using pseudo-tensor categories. As applications, we show that infinitesimal deformations and abelian extensions are classified by the second cohomology groups with different coefficients. Various examples are given to illustrate the difference between the cohomology of a Novikov algebra and that of the underlying pre-Lie algebra.

We investigate a special ansats that allows for an iterative solution of the constant Yang-Baxter equation. Testing this ansatz, we construct four sequences of the constant R-matrices. In each sequence the R-matrices act on the tensor squares of vector spaces of linearly growing dimensions. Each R-matrix also depends on a single complex parameter. By analyzing the spectra of the R-matrices, we conclude that the first and the third series are associated with the symmetric tensor representations of the quantum groups U_q(sl(2)) and U_q(sl(2|1)), respectively. The other two series appear to be related to representations of quantum groups in the case where q is a root of unity. The elements of the second series are associated with nilpotent representations of U_q(sl(2)). We also check the correspondence between these R-matrices and link invariants. Applying the Reshetikhin-Turaev procedure to the first, second, and fourth sequences yields, respectively, colored Jones polynomials, ADO invariants, and Alexander polynomials. To our surprise, a similar calculation for the third series yields a trivial result, identically equal to 1.

Retrieval-augmented generation is usually formulated as a statistical information-retrieval problem. Graph-based variants add relational structure, but the mathematical status of that structure is often left underspecified. Three distinct questions tend to be conflated: which statements are certified as knowledge, which latent representations are useful for retrieval, and which multi-hop compositions are semantically admissible. We propose Proof-Carrying Algebraic-Geometric Retrieval (PAGR), a framework that separates these questions mathematically. Its symbolic layer is a many-sorted relational theory generated by a typed quiver, path equations, and positive Horn inclusions. A quiver representation assigns inner-product spaces to entity types and linear operators to relations. A cellular sheaf measures local-to-global consistency. Semiring provenance records derivations and supports machine-checkable certificates. The central principle is epistemic separation: learned geometry may rank and organize evidence, but cannot promote a hypothesis to certified ground truth. We show the certification criterion is invariant under arbitrary replacement of learned components. Further results include a conditional completeness bound, identification of the isometry group as the relevant symmetry for residual-based retrieval, a cohomological consistency diagnostic, and a bounded-bisimulation index for admissible-path expansion. PAGR is a mathematical architecture for separating where a system should look from what it is allowed to treat as knowledge.

We record two results on additive diameters of group representations. First we give a corrected version of Proposition 5.3 from the first arXiv version of arXiv:2504.07573. Let $0<\varepsilon<1/3$, let $n>9/\varepsilon^2$, and let $U\leq\mathfrak{sl}_n(\mathbb{C})$ with $\dim U>\varepsilon n^2$. Then $\operatorname{diam}^{\mathrm{SL}_n(\mathbb{C})}_{+}(\mathfrak{sl}_n(\mathbb{C}),U)\leq 32/\varepsilon+8$. The proof uses an averaging argument for the action of the symmetric group on the off-diagonal matrix positions. This argument was subsequently developed further in arXiv:2609.03882. We also prove the right-hand inequality in Question 6.7 of arXiv:2504.07573. If $G$ is a complex algebraic group, $V$ is a finite-dimensional $G$-module, $U\leq V$, and $\mathfrak g=\operatorname{Lie}(G)$, then $\operatorname{diam}^{G}_{+}(V,U)\leq\operatorname{diam}^{\mathfrak g}_{+}(V,U)$. No irreducibility or connectedness assumption is needed.

Let $p,q$ be positive braids with Demazure products $u,v$. The endpoint stratum $\mathrm{Conf}(p,q)_{u,v}$ of the double Bott-Samelson cell splits as $\mathrm{Conf}(u,v) \times X(p) \times X(q^{\mathrm{op}})$, a double Bruhat cell times two braid varieties. We prove that the stratum's cluster structure is given by a cluster localization of the one on $\mathrm{Conf}(p,q)$, and that the splitting map is a quasi-cluster isomorphism. This comes from a mutation sequence along a double Demazure weave, one mutation per trivalent vertex, ending at an amalgamation of an extension of the double word quiver with the weave quiver. In the half-decorated case, this proves the conjecture of Gorsky-Kim-Scroggin-Simental for the splicing map $X(p) \times X(\Delta\Delta) \to X(p\Delta)$, where $\Delta$ is a reduced positive braid for $w_0$ and $\mathrm{dem}(p) = w_0$.

In this paper, we study thick subcategories of the (stable) module categories of finite-dimensional weakly symmetric algebras with radical cube zero. When the spectral radius of the Ext matrix is two, thick subcategories containing the algebra, equivalently thick subcategories of the stable category, are classified by Serre subcategories of a certain abelian category; when it is not two, only the trivial thick subcategories occur. As an application, this yields, to the best of our knowledge, the first examples of commutative Noetherian Gorenstein dominant local rings that are not complete intersections.

Over any local field of characteristic zero, we develop a local theory of Jacquet-Shalika integrals that were originally introduced by Jacquet and Shalika to study exterior square local $L$-factors for $GL_m$. In particular, at least in the archimedean case, we establish a complete functional equation matching Artin local $\varepsilon$-factors. Central to the argument is a novel comparison strategy between Jacquet-Shalika integrals and open-orbit integrals associated to the Shalika subgroup.

We explain a general procedure to compute characters of chiral algebras of 2d (S)CFTs geometrically from their coadjoint orbits, in analogy with the coadjoint orbit method for finite-dimensional Lie groups. We demonstrate that this method can be used to give concise rederivations of the characters of the Virasoro algebra, Kac-Moody algebras, as well as supersymmetric generalizations.

We show that the derived categories of smooth families of twisted generalized flag varieties admit semiorthogonal decompositions into derived categories of twisted sheaves over the base. In particular, we obtain a categorification of Panin's computation of the Quillen K-theory of twisted flag varieties. The main ingredient is a generalization of the Samokhin-van der Kallen semiorthogonal decomposition for representation categories of parabolic subgroups from split simply connected semisimple groups to arbitrary quasi-split semisimple groups, including non-simply connected cases.

Let $F$ be a non-archimedean local field of characteristic zero. Let $M$ be a Levi subgroup of $G = \mathrm{SL}(n,F)$. Consider the action on the discrete series of $M$ of the group of unitary unramified characters of $M$ (where a character acts by twisting). Given an orbit $\mathcal{O}$ for that action, let $W_\mathcal{O}$ be the global stabilizer of $\mathcal{O}$ in the Weyl group of $M$. We construct families of orbits $\mathcal{O}$ for which the action of $W_\mathcal{O}$ has no fixed point. More precisely, let $n \geq 8$ be an integer that is not prime and not equal to $9$. We attach families of orbits without fixed points to any divisor $m \geq 2$ of $n$ such that $m \;| \; q-1$ where $q$ is the order of the residue field of $F$. The Levi subgroup $M \subset G$ for the corresponding families has $m$ blocks of size $k\geq 2$ and $2$ blocks of size $m$. The core of the paper is the construction of superculpidal representations of $\mathrm{GL}(m,F)$ and $\mathrm{GL}(k,F)$ satisfying some conditions that we require to construct the families of orbits $\mathcal{O}$ that we expects. In addition, we study the fixed point problem for the dimensions $n$ that we excluded above, i.e., prime or small dimensions. We prove that there is always a fixed point under the action of $W_\mathcal{O}$ in those cases.

Using the language of q-opers, we propose an algebro-geometric description of the duality between Bethe Ansatz equations for quantum XXZ spin chains and many-body trigonometric Ruijsenaars-Schneider/Macdonald systems for classical groups. Upon this duality, the energy level sets of the classical B/C/D-type Hamiltonians are found to be in bijection with the set of solutions of the XXZ Bethe Ansatz equations of type A, albeit with open boundary conditions. This dictionary generalizes previous results in which a GL(N) XXZ spin chain with twisted periodic boundary conditions was dual to an N-body Ruijsenaars-Schneider system. Our construction implements a $\mathbb{Z}/2\mathbb{Z}$ folding both at the level of the (GL(N),q)-oper data as well as at the level of the N-body trigonometric Ruijsenaars-Schneider model. The nonreduced BC-type systems appear in our analysis as well.

Let $G=\mathrm{SL}_n(\mathbf C)$ and let $P\subset G$ be a standard parabolic subgroup with Levi factor $L$. For a $G$-dominant weight $\lambda$, consider the vector bundle on $T^*(G/P)$ obtained by pulling back the vector bundle on $G/P$ associated to the irreducible $L$-module $V_L(\lambda)^*$. We express its cohomology as a direct limit of the cohomology of certain line bundles on a Bott–Samelson variety associated to an affine Kac-Moody group. This comparison yields vanishing of higher cohomology and shows that the global sections are generated over $\mathbf C[\mathfrak g^*]$ by their degree-zero part $V_G(\lambda)^*$. Using these results together with the Braverman–Kazhdan intertwiners constructed in earlier joint work with A. Slipper, we give an explicit generating set for $\mathbf C[T^*(\mathrm{SL}_n/[P,P])]$. Finally, in an appendix joint with Tom Gannon, we combine these results to show the affinization $\mathrm{Spec}(\mathbf{C}[T^*(SL_n/[P,P])])$ has terminal singularities.

We write down generators and relations for a minimalist version of the nilHecke category which allows us to categorify the divided powers of a generating functor. We then classify these minimalist counterparts of the nilHecke category, subject to suitable assumptions. A minimalist version of the odd nilHecke category is introduced as well.

A finite simple graph $G$ is called a cograph if it does not contain the path on four vertices $P_4$ as an induced subgraph. It is classically known that the family of cographs are well-quasi-ordered by the induced subgraph relation \cite{D}. In preceding work of Knudsen and the third author \cite[Theorem 7.2]{KR}, it was shown that this well-quasi-order statement admitted a categorification, which allowed those authors to prove universal finite generation statements about the homology groups of configuration spaces on cographs \cite[Theorem 1.5]{KR}. In this work, we expand \cite[Theorem 7.2]{KR} to be compatible with the family of polynomial rings on the vertex sets of cographs. By consequence, we are able to prove a number of universality results related with edge and toric ideals of these polynomial ring, partially generalizing and expanding upon work of Kahle \cite{kahle2019binomial}. We also conclude strong restrictions on the kinds of topologies that can arise from graph complexes and anchored configuration spaces associated to cographs, as well as combinatorial constraints on the possible combinatorics of hyperplane arrangements of cographs.

Let $$X_n=\sum_{i=1}^s t_i A_i^{\otimes n},$$ where $A_1,\ldots,A_s\in M_d(\mathbb C)$ and $t_1,\ldots,t_s\in\mathbb C$ are fixed, while $n$ grows. Direct computation of determinants or Schatten norms of $X_n$ is exponential in $n$. For a single tensor power these quantities are elementary, and even the determinant of a generic two-term combination admits a reduction to polynomially many scalar factors; however, no analogous elementary reduction is available for three or more terms. We give an exact representation-theoretic method which, for fixed $d$ and $s$, computes $\|X_n\|_p$, $0<p<\infty$, and determinants in polynomial time in $n$. Schur–Weyl duality yields a simultaneous block decomposition, while Jacobi–Trudi identities in the Grothendieck ring replace Schur modules by signed combinations of tensor products of symmetric powers. For $d=3$, each irreducible contribution reduces to the difference of two explicitly computable symmetric-power terms, leading to an open-source implementation. In a single-thread CPU benchmark, a genuine three-term $3\times3$ trace-norm problem with $n=18$ is evaluated in about $47$ seconds, whereas just storing the unreduced matrix would require approximately $2.4\times10^{18}$ bytes. Direct and reduced computations agree to relative error below $3.4\times10^{-15}$ throughout their common range $n\leq9$.

This dissertation studies Cartan representations: representations of certain Coxeter groups as the Weyl groups of Kac-Moody algebras. It sets out and proves conditions for these representations to be isomorphic to each other, the reflection representation of the Coxeter group, and their dual representations, and gives partial results for the same over fields of finite characteristic, as well as clarifying what it means for a Coxeter group to be crystallographic and giving conditions for the Weyl group of a Kac-Moody algebra to be such. Some illustrative examples are also provided.

We define the new basis of the Grothendieck group of unipotent representations in an exceptional Weyl group, making more precise and correcting results in previous papers.

We prove a necessary condition conjecture ([27, Conjecture B]) for the maximal singularity of the Hilbert scheme of points in 3D for a large class of degrees, which in particular implies the 1978 Briançon-Iarrobino Conjecture for a tetrahedral degree, and vastly generalises the result in [19]. The novelty lies in introducing a new upper bound on the dimensions of the tangent spaces of the claimed non-maximally singular ideals, constructing canonical ideals that satisfy the claimed maximal singularity condition, providing a lower bound on the dimension of their tangent spaces, and finally comparing the two cases.

Let $p$ be an odd prime and let $\mathbf{G}/\mathbb Z$ be a split connected reductive group with $\operatorname{dim} Z(\mathbf{G})\leq1$. Assume that a split maximal torus of $\mathbf{G}$ admits a cocharacter whose pairing with every simple root is odd, and impose an explicit root-theoretic condition on the reduction of $\mathrm{Lie}(\mathbf{G}^{\mathrm{der}})$ modulo $p$. We construct infinitely many continuous representations $\rho:G_{\{p\}}\longrightarrow\mathbf{G}(\mathbb{Z}_p)$ which are unramified at every finite prime different from $p$ and whose images contain a principal congruence subgroup of $\mathbf G^{\mathrm{der}}(\mathbb{Z}_p)$. When the center has dimension one, the representations may be chosen to have open image in $\mathbf{G}(\mathbb{Z}_p)$. The construction continues the author's earlier work on $\mathrm{GL}_n$-valued representations ramified at one prime, but replaces the residual unobstructedness used there by a relative lifting argument. For $\mathbf{G}=\mathrm{GL}_n$ all the required root-theoretic conditions are automatic for every odd prime. We therefore obtain, for every odd $p$ and every $n>1$, infinitely many representations $G_{\{p\}}\to\mathrm{GL}_n(\mathbb{Z}_p)$ with open image. In particular, this removes the weak Vandiver-type hypothesis occurring in the author's previous work, as well as the restriction $p\geq7$ in that construction.

Let a group $G$ act linearly on a finite-dimensional complex vector space $V$. The group-additive diameter of a subspace $U \leq V$ is the least number of translates of $U$ whose sum is all of $V$. Counting dimensions, it is at least $\dim V / \dim U$. We show that when $G$ is compact and $V$ is irreducible, the diameter of every nonzero subspace is at most $\lceil (\dim V / \dim U) \ln \dim V \rceil$, so the trivial bound is correct up to a logarithmic factor. We measure the discrepancy by the covering complexity $\mathsf{C}(V)$, the largest ratio between the diameter of any subspace and its trivial lower bound, so that $1 \leq \mathsf{C}(V) \leq 2 + \ln \dim V$, and we determine where in this range various representations lie. Every irreducible representation of $\mathrm{SL}_2(\mathbf{C})$ has $\mathsf{C}(V) = 1$. The logarithm can be genuinely present along families of symmetric and exterior powers of $\mathrm{SL}_n(\mathbf{C})$ with $n$ varying, and it is present for finite Heisenberg groups and $2$-transitive groups of small order such as $\mathrm{PSL}_2(\mathbf{F}_p)$. The complexity is bounded above by a constant on the conjugation representations of $\mathrm{SL}_n(\mathbf{C})$ and on the representations $\operatorname{Sym}^k \mathbf{C}^3$ of $\mathrm{SL}_3(\mathbf{C})$. On the other hand, every fixed connected reductive group has a family of irreducible representations whose complexities tend to at least the dimension of the flag variety. Finally, for the Lie algebra $\mathfrak{sl}_3(\mathbf{C})$ acting on $\operatorname{Sym}^k \mathbf{C}^3$, the monomial diameter with respect to $\operatorname{Sym}^k X$ for a plane $X \leq \mathbf{C}^3$ is optimal, while the corresponding $\mathrm{SL}_3(\mathbf{C})$ diameter is not.

We prove a Haglund–Haiman–Loehr type combinatorial formula for the torus fixed point classes $I_{\mu,w}$ in the equivariant $K$-theory $K_{\mathbb{C}^{*}\times\mathbb{C}^{*}}(\operatorname{PFH}_{n,n-k})$ of parabolic flag Hilbert schemes. Under the identification of Bechtloff Weising and Orr, our result gives an HHL type formula for modified partially symmetric Macdonald functions in terms of weighted partial Dyck paths. When $w=\emptyset$, it specializes to the classical HHL formula. The proof relies essentially on the Carlsson–Gorsky–Mellit action of the Carlsson–Mellit algebra $\mathbb{A}_{q,t}$.

We prove a positivity theorem for the universal Plücker coordinates of Karp and Purbhoo when exactly one parameter is negative, with a sharp uniform threshold governed by the largest Plancherel up-transition probability. At the critical specialisation, positivity of the normalised $(2,2)$-coordinate is equivalent to the Octopus inequality, the main technical tool in the proof of Aldous's spectral gap conjecture. By connecting the inequality with Schubert calculus, this answers questions raised by Caputo and Aldous. For a Wronskian with distinct real zeros, we determine the maximal common half-line on which all branching-balanced coordinates are positive semidefinite. We also give Plücker-theoretic proofs of two hypergraph inequalities of Alon, Kozma, and Puder.

Granville and Ono proved that there is an $e$-core partition of $n$ for every $n\in\mathbb{N}$ if and only if $e\geq 4$, which translates to a statement about the existence of defect $0$ blocks for symmetric groups in positive characteristic and defect $0$ unipotent blocks of finite general linear groups in positive, non-defining characteristic. Motivated by analogous applications to the block theory of finite classical groups, imprimitive spetses, and cyclotomic Hecke algebras, we prove similar positivity statements about different variants of $e$-cores for charged multipartitions.

We prove the Jantzen conjecture for standard modules of graded affine Hecke algebras with equal parameters, real central character in the dominant deformation direction. The Jantzen layers are shown to be semisimple, and the multiplicities are given by the full local intersection cohomology of the corresponding orbit closures. The proof identifies the algebraic standard-to-contragredient map with the costalk-to-stalk map on a contracting transverse slice. Hard Lefschetz theorem determines its elementary divisors, and a residual grading on the specialized convolution algebra proves semisimplicity. An example in type $A$ is included to show that dominance is necessary.

A Hovey triple $(\mathcal{C}, \mathcal{F}, \mathcal{W})$ is exceptional, if $(\mathcal{C} \cap \mathcal{F}, \ \mathcal{C} \cap \mathcal{F} \cap \mathcal{W})$ is not a Frobenius pair. One has a disjoint union $\{\text{Hovey triple}\} = \{\text{Hereditary Hovey triple}\} \ \dot\bigcup \ \{\text{Exceptional Hovey triple}\}$ \ $\dot\bigcup \ \{\text{non-hereditary and non-exceptional Hovey triple}\}.$ Exceptional Hovey triples appear widely. For a selfinjective Nakayama algebra $A= kC_n/J^t$, $A\mbox{-}{\rm mod}$ admits exceptional Hovey triples if and only if $\gcd (n, t) \ge 2$ and $t \geq 3$. Their homotopy categories reveal new phenomena. Nakaoka-Palu's Theorem implies that there is a triangulation on $\frac{\mathcal C\cap\mathcal F}{\mathcal C\cap\mathcal F\cap\mathcal W}.$ It is proved that a Hovey triple in a weakly idempotent complete extriangulated category $(\mathcal A, \mathbb E, \mathfrak s)$ is exceptional if and only if the induced extriangulated structure $\bigl(\frac{\mathcal C\cap\mathcal F}{\mathcal C\cap\mathcal F\cap\mathcal W}, \ \overline{\mathbb E}, \ \overline{\mathfrak s}\bigr)$ is not canonically triangulated. Between this extriangulated structure and the one arising from the triangulation, the intermediate extriangulated structures are in one-to-one correspondence with Serre subcategories of $\mathrm{fp}((\frac{\mathcal P(\mathcal C\cap\mathcal F)}{\mathcal C\cap\mathcal F\cap\mathcal W})^{\mathrm{op}}, \mathrm{Ab})$.

We modify the definition of eventually homological isomorphisms so that it becomes more natural and fruitful. We introduce singular equivalences of $n$-adjoint type which can be viewed as standard singular equivalences of different levels and are more convenient for reducing various homological conjectures and transferring homological properties. We develop the theories of singular equivalences of $n$-adjoint type for $n\ge 2$ and standard eventually homological isomorphisms between derived categories by giving their characterizations in terms of $\otimes$-(co)perfect complexes of bimodules, two constructions via opposite algebras and tensor product algebras, examples induced by homomorphisms of algebras and idempotents, and applications to the reduction of homological conjectures such as the finitistic dimension conjecture, Gorenstein symmetry conjecture, Auslander-Reiten conjecture and Gorenstein projective conjecture, as well as to the preservation of homological properties such as Gorensteinness, the Fg condition, Happel's property and Han's property. Moreover, we show that standard eventually homological isomorphisms correspond to fully faithful standard triangle functors between singularity categories with canonical left adjoint, and essentially surjective standard eventually homological isomorphisms between derived categories correspond to singular equivalences of 3-adjoint type, thereby incorporating essentially surjective standard eventually homological isomorphisms into the framework of singular equivalences of 3-adjoint type.

We use representation theory to calculate the $\pi$-electron energy levels of Buckminsterfullerene and other symmetrical conjugated systems according to the Hückel approximation.

Smoothness of a function on the real line is reflected in the decay of its Fourier transform, which suggests that smoothness of a function in $L^2(G)$ for a group $G$ should mean concentration of the Fourier coefficients at low frequency. Such a reading presupposes an ordering of the irreducible representations of $G$, but for non-abelian $G$, no ordering is canonical. Given a symmetric generating set $S$, the Laplacian of the associated Cayley graph is block diagonal over the dual, and we order the irreps by the mean of the eigenvalues in each block. This produces an ordering function $\omega:\widehat{G}\to\mathbb{R}$ that depends only on the pair $(G,S)$. This function is bounded between zero and two, vanishing only at the trivial representation and achieving the upper bound exactly when the Cayley graph is bipartite. We then ask how much freedom the construction has. Within the class of operators satisfying natural axioms, the induced orderings are exactly the real functions on the dual vanishing at the trivial representation and agreeing on conjugate pairs, and the orderings coming from inversion orbits of conjugacy classes form a basis for them. We cut the freedom down further by requiring two additional inputs: nonnegativity of the class weights and a declaration of which group elements count as uniform incremental changes, which pins the operator to the Cayley-Laplacian up to positive scale. We observe that the construction persists for compact groups even though the Cayley graph does not, and we extend the theory to finite sets carrying a transitive group action, where the acting group selects which frequencies exist and the generating set orders them. The answer to the title question is therefore that smoothness is a property of a function together with a choice of group and generating set, not of the function alone.

Using a bijection between the lattice points in a lecture hall polytope and Lakshimibai-Seshadri (L-S) paths, we prove the Koszul property of lecture hall polytopes in complete generality, and give new proofs for their Integral Decomposition Property and for a criterion on Gorenstein property.

For a given function $f$ on a discrete subspace $\Gamma$ of a noncompact Riemannian symmetric space $G/K$, we construct analytic, noncompactly supported functions $W$ on $G/K$ with $W|_{\Gamma}=f$ that, in addition, satisfy an invariant differential equation on $G/K$. The key step is to regard $W$ as an orthogonal projection onto a space of coherent states in a representation of the Lie group $G$ in a Hilbert space that admits a strictly positive definite reproducing kernel. The techniques are implemented for the heat kernel on $G/K$, and $W$ is expressed in terms of certain closed formulas for linearly independent coherent states.