CyberSec Research

Given a Sylow $p$-subgroup $P$ of a symmetric group, we describe the action of its normalizer on $\mathrm{Irr}(P)$. To this end, we establish a one-to-one correspondence between the irreducible characters of $P$ and certain equivalence classes of explicitly defined functions, which are also naturally suited to describing the Galois action.

Under a largeness assumption on the size of the residue field, we give an explicit description of the positive-depth Deligne--Lusztig induction of unramified elliptic pairs $(T,\theta)$. When $\theta$ is regular, we show that positive-depth Deligne--Lusztig induction gives a geometric realization of Kaletha's Howe-unramified regular $L$-packets. This is obtained as an immediate corollary of a very simple "litmus test" characterization theorem which we foresee will have interesting future applications to small-$p$ constructions. We next define and analyze Green functions of two different origins: Yu's construction (algebra) and positive-depth Deligne--Lusztig induction (geometry). Using this, we deduce a comparison result for arbitrary $\theta$ from the regular setting. As a further application of our comparison isomorphism, we prove the positive-depth Springer hypothesis in the $0$-toral setting and use it to give a geometric explanation for the appearance of orbital integrals in supercuspidal character formulae.

Given a finite-dimensional inner product space $V$ and a group $G$ of isometries, we consider the problem of embedding the orbit space $V/G$ into a Hilbert space in a way that preserves the quotient metric as well as possible. This inquiry is motivated by applications to invariant machine learning. We introduce several new theoretical tools before using them to tackle various fundamental instances of this problem.

The goal of our work is to study the decomposition of the joint action of $\mathfrak{g} = \mathfrak{spo}(2n|1)$ and $\mathfrak{g}' = \mathfrak{osp}(2|2)$ on the supersymmetric algebra S = S($\mathbb{C}^{2n|1} \otimes \mathbb{C}^{1|1}$). As proved by Merino and Salmasian, we have a one-to-one correspondence between irreducible representations of $\mathfrak{g}$ and $\mathfrak{g}'$ appearing as subrepresentations of S. In this paper, we obtained an explicit description of the highest weights and joint highest weight vectors for the representations of $\mathfrak{g}$ and $\mathfrak{g}'$ appearing in the duality.

Combining results from Keller and Buchweitz, we describe the 1-periodic derived category of a finite dimensional algebra $A$ of finite global dimension as the stable category of maximal Cohen-Macaulay modules over some Gorenstein algebra $A^\ltimes$. In the case of gentle algebras, using the geometric model introduced by Opper, Plamondon and Schroll, we describe indecomposable objects in this category using homotopy classes of curves on a surface. In particular, we associate a family of indecompoable objects to each primitive closed curve. We then prove using results by Bondarenko and Drozd concerning a certain matrix problem, that this constitutes a complete description of indecomposable objects.

In this paper, we aim to study abelian extensions for some infinite group. We show that the Hopf algebra $\Bbbk^G{}^\tau\#_{\sigma}\Bbbk F$ constructed through abelian extensions of $\Bbbk F$ by $\Bbbk^G$ for some (infinite) group $F$ and finite group $G$ is cosemisimple, and discuss when it admits a compact quantum group structure if $\Bbbk$ is the field of complex numbers $\mathbb{C}.$ We also find all the simple $\Bbbk^G{}^\tau\#_{\sigma}\Bbbk F$-comodules and attempt to determine the Grothendieck ring of the category of finite-dimensional right $\Bbbk^G{}^\tau\#_{\sigma}\Bbbk F$-comodules. Moreover, some new properties are given and some new examples are constructed.

The present work is inspired by three interrelated themes: Weingarten calculus for integration over unitary groups, monotone Hurwitz numbers which enumerate certain factorisations of permutations into transpositions, and Jucys-Murphy elements in the symmetric group algebra. The authors and Moskovsky recently extended this picture to integration on complex Grassmannians, leading to a deformation of the monotone Hurwitz numbers to polynomials that are conjectured to satisfy remarkable interlacing phenomena. In this paper, we consider integration on the real Grassmannian $\mathrm{Gr}_\mathbb{R}(M,N)$, interpreted as the space of $N \times N$ idempotent real symmetric matrices of rank $M$. We show that in the regime of large $N$ and fixed $\frac{M}{N}$, such integrals have expansions whose coefficients are variants of monotone Hurwitz numbers that are polynomials in the parameter $t = 1 - \frac{N}{M}$. We define a "$b$-Weingarten calculus", without reference to underlying matrix integrals, that recovers the unitary case at $b = 0$ and the orthogonal case at $b = 1$. The $b$-monotone Hurwitz numbers, previously introduced by Bonzom, Chapuy and Dolega, arise naturally in this context as monotone factorisations of pair partitions. The $b$- and $t$-deformations can be combined to form a common generalisation, leading to the notion of $bt$-monotone Hurwitz numbers, for which we state several results and conjectures. Finally, we introduce certain linear operators inspired by the aforementioned $b$-Weingarten calculus that can be considered as $b$-deformations of the Jucys-Murphy elements in the symmetric group algebra. We make several conjectures regarding these operators that generalise known properties of the Jucys-Murphy elements and make a connection to the family of Jack symmetric functions.

In this paper, we investigate pre-Calabi-Yau algebras and homotopy double Poisson algebras arising from homotopy Rota-Baxter structures. We introduce the notion of cyclic homotopy Rota-Baxter algebras, a class of homotopy Rota-Baxter algebras endowed with additional cyclic symmetry, and present a construction of such structures via a process called cyclic completion. We further introduce the concept of interactive pairs, consisting of two differential graded algebras-designated as the acting algebra and the base algebra-interacting through compatible module structures. We prove that if the acting algebra carries a suitable cyclic homotopy Rota-Baxter structure, then the base algebra inherits a natural pre-Calabi-Yau structure. Using the correspondence established by Fernandez and Herscovich between pre-Calabi-Yau algebras and homotopy double Poisson algebras, we describe the resulting homotopy Poisson structure on the base algebra in terms of homotopy Rota-Baxter algebra structure. In particular, we show that a module over an ultracyclic (resp. cyclic) homotopy Rota-Baxter algebra admits a (resp. cyclic) homotopy double Lie algebra structure.

This paper classifies complex local representations of the twisted virtual braid group, $\mathrm{TVB}_2$, into $\mathrm{GL}_3(\mathbb{C})$. It shows that such representations fall into eight types, all of which are unfaithful and reducible to degree $2 \times 2$. Further reducibility to degree 1 is analyzed for specific types. The paper also examines complex homogeneous local representations of $\mathrm{TVB}_n$ into $\mathrm{GL}_{n+1}(\mathbb{C})$ for $n \geq 3$, identifying seven unfaithful types. Additionally, complex local representations of the singular twisted virtual braid group, $\mathrm{STVB}_2$, into $M_3(\mathbb{C})$ are classified into thirteen unfaithful types. Finally, the paper demonstrates that not all complex local extensions of $\mathrm{TVB}_2$ representations to $\mathrm{STVB}_2$ conform to a $\Phi$-type extension.

We show that Kessar's isotypy between Galois conjugate blocks of finite group algebras does not always lift to a $p$-permutation equivalence. We also provide examples of Galois conjugate blocks which are isotypic but not $p$-permutation equivalent. These results help to clarify the distinction between a $p$-permutation equivalence and an isotypy, and may be useful in determining necessary and sufficient conditions for when an isotypy lifts to a $p$-permutation equivalence.

This paper explores persistence modules for circle-valued functions, presenting a new extension of the interleaving and bottleneck distances in this setting. We propose a natural generalisation of barcodes in terms of arcs on a geometric model associated to the derived category of quiver representations. The main result is an isometry theorem that establishes an equivalence between the interleaving distance and the bottleneck distance for circle-valued persistence modules.

In recent decades, the structure of the mod-2 cohomology of the Steenrod ring $\mathscr A$ has become a major subject of study in the field of Algebraic Topology. One of the earliest attempts to study this cohomology through the use of modular representations of the general linear groups was the groundbreaking work [Math. Z. 202 (1989), 493-523] by W.M. Singer. In that work, Singer introduced a homomorphism, commonly referred to as the "algebraic transfer," which maps from the coinvariants of a certain representation of the general linear group to the mod-2 cohomology group of the ring $\mathscr A.$ Singer's conjecture, in particular, which states that the algebraic transfer is a monomorphism for all homological degrees, remains a highly significant and unresolved problem in Algebraic Topology. In this research, we take a major stride toward resolving the Singer conjecture by establishing its truth for the homological degree four.

We find equivalent conditions determining the representation type of abelian restricted Lie algebras in terms of how their Green ring of restricted representations varies with respect to different cocommutative Hopf algebra structures on its restricted universal enveloping algebra. Each compatible cocommutative Hopf algebra structure on a tame algebra is shown to have a correspondence between a certain set of Hopf subalgebras, and the set of minimal thick tensor-ideals having identical ring structure when determined by either the Hopf algebra structure or the base Lie algebra structure (up to a choice of character group). Those of wild representation type are shown never to have such a correspondence.

Twisted loop algebras of the second kind are infinite-dimensional Lie algebras that are constructed from a semisimple Lie algebra and an automorphism on it of order at most $2$. They are examples of equivariant map algebras. The finite-dimensional irreducible representations of an arbitrary equivariant map algebra have been classified by Neher--Savage--Senesi. In this paper, we classify the finite-dimensional irreducible representations of twisted loop algebras of the second kind in a more elementary way.

We investigate the relationship between the delooping level (dell) and the finitistic dimension of left and right serial path algebras. These 2-syzygy finite algebras have finite delooping level, and it can be calculated with an easy and finite algorithm. When the algebra is right serial, its right finitistic dimension is equal to its left delooping level. When the algebra is left serial, the above equality only holds under certain conditions. We provide examples to demonstrate this and include discussions on the sub-derived (subddell) and derived delooping level (ddell). Both subddell and ddell are improvements of the delooping level. We motivate their definitions and showcase how they can behave better than the delooping level in certain situations throughout the paper.

Given an idempotent complete additive category, we show the there is an explicitly constructed topological space such that the lattice of exact substructures is anti-isomorphic to the lattice of closed subsets. In the special case that the additive category has weak cokernels, this topological space is an open subset of the Ziegler spectrum and this is a result of Kevin Schlegel. We also look at some module categories of rings where the Ziegler spectrum is known and calculate the global dimensions of the corresponding exact substructures.

We introduce more general notions of Clifford codes and stabilizer codes, the latter we call weak stabilizer codes. This is all formulated in the language of projective representation theory of finite groups and we give a novel description of the detectable errors for a Clifford code. We also give examples of infinite families of non-stabilizer Clifford codes as well as examples of non-Clifford weak stabilizer codes. The latter of these types of examples is a class of codes that have not been studied in the same systematic framework as Clifford codes and stabilizer codes.

Motivated by constructions from applied topology, there has been recent interest in the homological algebra of linear representations of posets, specifically in homological algebra relative to non-standard exact structures. A prominent example of such exact structure is the spread exact structure, which is an exact structure on the category of representations of a fixed poset whose indecomposable projectives are the spread representations (that is, the indicator representations of convex and connected subsets). The spread-global dimension is known to be finite on finite posets, and unbounded on the collection of Cartesian products between two arbitrary finite total orders. It is conjectured in [AENY23] that the spread-global dimension is bounded on the collection of Cartesian products between a fixed, finite total order and an arbitrary finite total order. We give a positive answer to this conjecture, and, more generally, we prove that the spread-global dimension is bounded on the collection of Cartesian products between any fixed, finite poset and an arbitrary finite total order. In doing this, we also establish the existence of finite spread-resolutions for finitely presented representations of arbitrary upper semilattices.