Papers

We establish a Newton-type identity for the queer Lie superalgebra $\mathfrak q_N$, relating Sergeev's odd cyclic central elements to Nazarov's one-row Capelli elements. The identity is obtained by comparing Ivanov's generating function for factorial Schur $Q$-functions with the queer Perelomov-Popov product of Grigoryev and Nazarov. Its coefficient expansion yields a triangular change of generators between the odd cyclic and odd one-row families. In particular, the odd one-row Capelli elements generate the center, while the even one-row elements are redundant. In fixed rank, we derive determinantal relations and a generic reconstruction theorem. The basic cyclic Hankel determinant is identified with a resultant and factored into the failure-of-strong-typicality and shifted-resonance factors. After localization at this determinant, the center is generated by the first $2N$ odd cyclic elements; consequently, generic central characters are determined by their values on these elements.

Let $\mathcal O$ be a nonzero nilpotent orbit in a complex simple Lie algebra $\mathfrak g$. For $\mathfrak g$ of classical type, we classify the orbit closures $\overline{\mathcal O}$ that are isomorphic to $(T^*X)^{\mathrm{aff}}$ for a smooth quasi-affine variety $X$. In types $G_2$, $F_4$, and $E_8$, we show that no nonzero nilpotent orbit closure admits such a cotangent model.

We obtain arithmetic conditions which are satisfied by solvable groups admitting an abelian $\pi$-subgroup. First, we extend a previous characterization by removing the square-free hypothesis on some fixed irreducible character degree. We then show the same arithmetic conditions follow from a faithful action of an abelian $\pi$-subgroup on the $\pi'$-part of the Fitting subgroup, where $\pi$ is the set of prime divisors of the aforementioned character degree.

The $\Lambda$-distribution of a compact matrix group is an invariant in algebraic probability theory that was recently introduced to study zero distributions of function field $L$-functions. It is encoded by the $\sigma$-moment generating function, a generalization of the Molien series of classical invariant theory. In this work, we compute the $\sigma$-moment generating functions of finite matrix groups in many new cases. In particular, we compute the asymptotic $\Lambda$-distributions for the infinite families of Weyl reflection groups of types $B_n/C_n$ and $D_n$, complementing the previously known case of reflection groups of type $A_n$, i.e., symmetric groups. We also establish a general result relating the shapes of $\sigma$-moment generating functions to the distributions of associated classical random variables, explaining a previous ad hoc observation for independent Gaussians arising from traces of powers on compact classical groups.

We propose a refinement of the quantum Hikita conjecture of Kamnitzer, McBreen, and Proudfoot that bridges the representation theory of Coulomb branches with the enumerative geometry of Higgs branches. We also introduce a general framework for proving it, which we carry out for ADE quiver gauge theories with minuscule framings and for the gauge theory corresponding to the Jordan quiver. As an application, we use the resulting quantum Hikita isomorphisms to give a geometric description of graded traces on quantized Coulomb branches.

A two-parameter hierarchy of classes of finitely generated modules over an artin algebra is described by using the $\mho$-quiver. As an application, it is shown that the stable category of reflexive modules is equivalent to two other categories via the mutually quasi-inverse equivalences induced by $\Omega$ and $\mho$.

We give a geometric description of nilpotent representations of equioriented cyclic quivers over a field via an explicit bijection. Over a finite field, this yields formulas for the number of nilpotent representations and for the probability that a representation is nilpotent. We also give a refinement by rank. Over the Boolean semiring, we identify nilpotent semirepresentations with directed acyclic graphs, derive a recursive formula for their number, and determine the asymptotic decay rate of the probability of nilpotence.

Let $G$ be a finite group and let $p$ be a prime. If $P$ is a nonabelian Sylow $p$-subgroup of $G$ and $m(P)$ is the smallest non-linear irreducible character degree of $P$, we prove that there exists $\chi \in {\rm Irr}(G)$ in the principal $p$-block of $G$ such that $1<\chi(1)_p\le m(P)$, giving one inequality of the Eaton-Moretó conjecture for principal blocks. This, assuming Dade's Projective conjecture, implies the Eaton-Moretó conjecture for principal blocks.

Let Phi be an irreducible reduced crystallographic root system of rank r at least 2, let N = |Phi^+| be the number of positive roots, and let h be its Coxeter number. For the normalized Witten zeta function xi_Phi, we determine the first distinct pole below the leading pole 2/h. It is located at q_2(Phi) = (r-1)/(N-1) = 2(r-1)/(rh-2), is simple, and receives contributions precisely from the codimension-one faces of the dominant chamber. Its residue is zeta_R(q_2)/(N-1) times the sum of the corresponding wall periods, where zeta_R denotes the Riemann zeta function. These periods are finite and positive, so the residue is strictly negative. We also prove a Stokes relation for projective hyperplane-arrangement periods. Let A be an essential central real arrangement in R^n with weights lambda_H strictly between 0 and 1. Suppose that the sum of lambda_H over all H in A equals n, and that for every nonzero proper intersection flat X the sum of lambda_H over those H containing X is strictly less than the codimension of X. Then the vector of positive chamber periods lies in the kernel of the Varchenko matrix with weights exp(pi i lambda_H). Applying this relation, we evaluate the relevant wall periods in types F4 and D5. A two-orbit reduction and Dixon's 3F2(1) summation give a gamma-product evaluation in type F4, while a four-orbit reduction and Selberg's integral give a gamma-product evaluation of the complete wall sum in type D5. Consequently, we obtain exact formulas for the normalized and ordinary Witten zeta residues at 3/23 and 4/19.

We give results on the question of whether the intersection numbers of a primitive group association scheme determine it up to combinatorial isomorphism. For $G=\operatorname{PSL}(2,q)$, where $q$ is an odd prime power with $q=11$ or $q\ge 17$, or $q=2^f$ with $f\ge3$, we construct a Schur partition that is algebraically isomorphic to the partition of $G$ into conjugacy classes but not combinatorially isomorphic to it. Consequently, the corresponding primitive group association schemes are not determined up to combinatorial isomorphism by their intersection numbers; in particular, they are non-separable. For $\mathfrak A_6$ and $\mathfrak A_8$, we also explicitly construct Schur partitions that are algebraically isomorphic to the corresponding partitions into conjugacy classes but not combinatorially isomorphic to them.

We propose a definition of the quantized Coulomb branches of separated cotangent type, and prove that the corresponding classical construction recovers the non-cotangent Coulomb branch. We also obtain a formula for quasi-minuscule monopole operators in arbitrary cotangent type. Applying these results, we compute the monopole operators for orthosymplectic quivers and construct a homomorphism from the shifted twisted Yangian of split ADE type to the corresponding quantized Coulomb branch algebra.

For $\gcd(a,b)=1$, we show that the moduli space of $m$-codimensional $\Bbbk[\![T^a,T^b]\!]$-submodules of $\Bbbk[\![T]\!]^n$ is paved by affine cells, by proving that each Białynicki-Birula stratum of a closed related moduli space with respect to the natural $\mathbb{G}_m$-action is an affine bundle over the fixed point locus and that the fixed point locus is an iterated Grassmannian bundle. As an application, we determine the motive of this moduli space in the Grothendieck ring of varieties in terms of an explicit two-variable series $N_{a,b;n}(q,t)$, and use it to explicit compute the groupoid volume of the category of finite modules over $\mathbb{F}_q[\![T^a,T^b]\!]$. The series $N_{a,b;n}$ carries the conjectures we then formulate. At $n=\infty$ we conjecture a bi-infinite family of Rogers–Ramanujan type identities by specializing the $t$-variable; we identify their product side with the normalized character of a module over the $\mathcal{W}$-algebra minimal model $\mathcal{W}_a(a,a+b)$, and observe a connetion to colored Jones tails. At $n<\infty$ we conjecture that $N_{a,b;n}$ is computed by the bottom $\alpha$-row of the trigraded $S^n$-colored HOMFLY homology of the torus knot $T(a,b)$, and that this same bottom row also computes the Quot schemes of finite codimensional $\Bbbk[\![T^a,T^b]\!]$-submoudles of $\Bbbk[\![T^a,T^b]\!]^n$ and the punctual Hilbert schemes of the non-reduced curve $(Y^a-X^b)^n=0$; the three quantities are special values at three points of the trigrading, and when $n=1$ they recover both the conjectures of Oblomkov–Rasmussen–Shende and of Kivinen–Trinh. Finally we conjecture that the one direction of the trigrading these three points do not see is a perverse filtration on the moduli spaces themselves, and we verify its prediction for a smooth germ at $n=2$ by computing the decomposition theorem for the $\mathrm{GL}_2$ spectral-curve family.

Let $F=\mathbb{F}_q((u))((t))$ and $H=\mathrm{GL}_n(F)_L\times\mathrm{GL}_n(F)_R$. An ordered tuple $\alpha=(\alpha_1,\dots, \alpha_n) $ defines a cross-$K_2$ multiplier on the doubled Borel. Twisted equivariantization then yields an exact uniformly smooth categorical principal series $2$-representation of $H$. For unitary parameters, the $\mathbb R((X))$-measure gives a positive Hom-valued formal Hermitian inner product on the spherical part. These constructions provide two-dimensional categorical analogues of selected unramified structures.

We complete our work on $\mathsf{GL}_2$ over $\mathbb{Q}$ in the ramified setting for Beyond Endoscopy proposed by Langlands. We prove that the asymptotic formula for each term of the trace formula when summing over $n<X$ with arbitrary smooth test functions at places in $S=\{\infty,q_1,\dots q_r\}$ with $2\in S$, for the standard representation, is $o(X)$. We prove an identity with a variable $X$, called the limit form of the trace formula for $\mathsf{GL}_2$ over $\mathbb{Q}$, directly. The proof uses Arthur's result on the Fourier transform of weighted orbital integrals to rewrite the term involving intertwining operators, and then compares the expansion with the results of the real case due to Arthur-Herb-Sally and Hoffmann, and the nonarchimedean case by direct computation using Arthur's definition.

We combine results of Rickard and Shaul with methods of intrinsic homological algebra and the theory of purity to prove that the finiteness of the big finitistic dimension of an Artin algebra is an intrinsic property of the singularity category of its opposite algebra, via its category of modules. This `object-free' approach complements work of Dey–Šťovíček and arrives at the same conclusion: the finiteness of the finitistic dimension of Artin algebras is a singular invariant (of the opposite algebras). In fact, our characterisation can be used to prove that finite finitistic dimension descends along certain fully faithful functors between singularity categories. We include an appendix where we explain how our methods can be used to prove non-existence of bounded t-structures on singularity categories.

In this note, we prove a recent conjecture of Hung, Martínez and Navarro on the sums of character degrees of finite groups.

We propose a categorification of the Lusztig integral form of the quantum Cartan subalgebra for $sl(2)$ via a colimit of categories of equivariant coherent sheaves on finite-dimensional projective spaces. The nonequivariant version of our construction yields a nonsemisimple abelian categorification of the ring of integer-valued polynomials.

The Rogers–Ramanujan identities equate a $q$-series whose exponents are governed by a quadratic form with an infinite product supported on two residue classes modulo~$5$. Identities of this shape are scarce, and a central problem is to identify the structures that produce them in families. Huang, Jiang, and Oblomkov have proposed a source of a new kind: to each pair of coprime integers $a,b>1$ they attach an infinite-rank $q$-series $Z_{a,b}(q)$, assembled from counts of commuting nilpotent matrix pairs $(A,B)$ with $A^a=B^b$ over finite fields, and they conjecture that it equals an explicit product of $(a-1)(b-1)/2$ modular units of level $a+b$. The $a=2$ cases are the Andrews–Gordon identities; no case with $a>2$ was known. We prove the conjecture for $(a,b)=(3,4)$, $(3,5)$, $(3,7)$, and $(3,8)$. Our proofs pass through a finer sum-to-sum identity, which we conjecture for all $b$ coprime to $3$ and establish for all $b$ when $q=1$. Lau and Ono have since proved that identity in general, and with it the full $a=3$ case. These identities have been formalized and verified in Lean by AxiomProver.

We show that, over any algebraically closed field of any characteristic, the category of finite-dimensional projective modules over the preprojective algebra of generalized Dynkin type $\mathbb{L}_2$ has a pretriangulated category structure which is neither an algebraic nor a topological triangulated category structure.

In this note we give an alternative derivation of Frobenius Character Formula using wedge product, the resulting formula is used to calculate the character table of $S_{2}$ and $S_{3}$ Group which reproduces their character table correctly, calculations for classes of $S_{3}$ Group are given in the appendix B. The calculations are given in terms Schur Functions. In the first section, a brief indicative review of Symmetric Group and related formulas is given. In Section second we derive the formula. A naive approach to computational complexity is also presented at the end.

In 1991 Rickard asked whether every derived equivalence between finite-dimensional algebras over a common field is standard. Hu, Xi and Zhang constructed non-standard derived equivalences over the field with two elements and conjectured that Rickard's question has a positive answer in characteristic different from two. We disprove this conjecture by constructing non-standard derived autoequivalences over every field. In fact, each field admits infinitely many finite-dimensional algebras with such autoequivalences. The proof is characteristic-free and is based on a supertrace identity for matrices over truncated polynomial algebras. We also give a second family over finite fields. Consequently, Rickard's question has a negative answer over every field.

In this paper, when the quiver Hecke algebra R is symmetric, we present a new construction of quantum monoidal seeds $ \mathscr{S}_{w,v}$ for $\mathcal{C}_{wv}$ using the reflection functors $\mathcal{F}_i$ and the newly introduced operators $\mathcal{K}_i$. The monoidal seed $ \mathscr{S}_{w,v}$ is obtained as a subseed of the monoidal seed of $\mathcal{C}_w$ constructed by applying $\mathcal{K}_i$ and $\mathcal{F}_i$ along the special KF sequence determined by a reduced expression of $w$ and $v$. We further prove that the monoidal seed $\mathscr{S}_{w,v}$ coincides with the set of all prime factors of the determinantial modules $M(w_{\le k } \Lambda_{i_k}, v_{\le k} \Lambda_{i_k} )$. We prove that the Grothendieck ring $K(\mathcal{C}_{wv}) $ lies between the cluster algebra and the upper cluster algebra.

Let $V$ be an $F_2$-vector space with a given nondegenerate symplectic form and a circular basis. We define a partition of $V$ into subsets each of which has cardinal equal to a product of Catalan numbers. This can be viewed as a partition of the set of unipotent representations of a finite symplectic group corresponding to a certain two-sided cell.

Given a pair of (round) 2-dimensional spheres, one of which has radius three times that of the other, the Lie algebra of local infinitesimal symmetries for the distribution corresponding to rolling the spheres along each other (without letting them slip or twist) happens to be isomorphic to the split real form of the exceptional simple Lie algebra of rank 2. These exceptional local symmetries appear only for this specific 1:3 ratio of radii, however, and while there are several proofs of this result, a straightforward kinematic explanation for the seemingly miraculous appearance of an exceptional simple Lie group in this situation has long been desired. In this paper, we provide such an explanation, relating the ratio of radii to the intersections of a pair of one-parameter subgroups that can be seen from the rolling sphere perspective. The approach does not require the split-octonions, as we construct the exceptional simple Lie group directly by Tanaka prolongation to aid in visualizing the underlying geometry.

Let $G$ be a connected solvable algebraic group acting rationally on a finite-dimensional vector space $U$. Using a $G$-stable complete flag, we formulate a successive-quotient procedure that decides whether $U$ contains an open $G$-orbit. When the procedure succeeds, it constructs an open-orbit vector of minimum support cardinality and determines the image of a generic stabilizer in the maximal torus quotient. We also give an infinitesimal version detecting open separable orbits. We apply the criterion to the action of $Aut_Q(M)$ on $Ext^1_Q(M,M)^*$ where $M$ is a multiplicity-free representation of a Dynkin quiver. Rigidity of the corresponding component of Lusztig's nilpotent variety is thereby reduced to a rank test together with an acyclicity condition on a graph of active extension coordinates; the connected components of the resulting forest determine the generic indecomposable decomposition. For equioriented type $A$ this yields an explicit algorithm for a family of multisegments encoded by incidence matrices, including nonregular examples with repeated beginnings or ends.

A geometric model for string algebras was recently established in \cite{BC24}. Building upon this framework, we characterize the class of string algebras whose geometric models are unique up to equivalence of labelled tiled surfaces. Moreover, we provide a necessary and sufficient condition for all geometric models of a string algebra to be entirely free of red punctures, and further give a combinatorial description of string algebras with a common red puncture across all geometric models. In addition, we derive a sufficient condition for a string algebra guaranteeing the presence of red punctures in all geometric models.

We introduce and study the category of unipotent character sheaves on the non-degenerate locus of the Vinberg semigroup associated to a reductive group of adjoint type. We show that appropriate perverse objects in this category can be glued from perverse parabolic character sheaves on the strata. As an application, we study the intermediate extensions of simple unipotent character sheaves from the group to its wonderful compactification, proving a conjecture of He. We also disprove a conjecture of Lusztig regarding the intermediate extension of the Steinberg character sheaf.

Fix nonzero $r_0,r_1\in\mathbb{C}$. Let $\widetilde{\mathcal H}$ denote a nil-DAHA of type $(C_1^\vee,C_1)$ defined by generators $t_0,u_0,t_1,u_1$ and relations $(t_i-r_i)(t_i-r_i^{-1})=0$ for $i\in\{0,1\}$, $u_0^2=u_0$, $u_1^2=0$, and $u_0t_0t_1u_1=0=t_1u_1u_0t_0$. Set $A=u_0t_0$ and $B=t_1u_1$. A finite-dimensional $\widetilde{\mathcal H}$-module is called $(A,B)$-multiplicity-free, or simply multiplicity-free, if $A$ and $B$ are simultaneously diagonalizable and every nonzero common eigenspace is one-dimensional. We consider finite-dimensional irreducible multiplicity-free modules that have a certain ordered basis, which we call an adapted block basis. For $D\geq 1$, we construct a family of $2D$-dimensional $\widetilde{\mathcal H}$-modules $E_D$, and for $D\geq 0$, we construct a family of $(2D+1)$-dimensional $\widetilde{\mathcal H}$-modules $O_D$. We determine which members of these families are multiplicity-free and irreducible. We prove that every finite-dimensional irreducible $\widetilde{\mathcal H}$-module that is multiplicity-free and has an adapted block basis is isomorphic to a module of the form $E_D$ or $O_D$. We also determine when two members of the same family are isomorphic.

This article studies dominant local rings, a notion introduced by Takahashi. We prove that, for a flat local homomorphism R–>S between commutative noetherian local rings, dominance descends from S to R, thereby answering Takahashi's descent question affirmatively. The proof makes use of Krause's realization of the idempotent completion of the singularity category as the subcategory of compact objects in the homotopy category of acyclic complexes of injective modules.

Let $R$ be a commutative noetherian ring. Denote by $\operatorname{mod} R$ the category of finitely generated $R$-modules. Let $\Delta$ be a subset of $\operatorname{Spec} R$, and let $\operatorname{Ass}^{-1}\Delta$ stand for the full subcategory of $\operatorname{mod} R$ consisting of finitely generated $R$-modules whose associated prime ideals belong to $\Delta$. In this paper, we consider classifying torsion pairs in $\operatorname{Ass}^{-1}\Delta$ and some other full subcategories of $\operatorname{mod} R$.