Linear Group Research Articles

On 2-distance primitive graphs of prime valency

A vertex transitive non-complete graph is called 2-distance primitive if the stabilizer of a vertex is primitive on both the first and the second step neighbourhoods. In 2019, Jin, Huang and Liu proved that all connected 2-distance primitive graphs of prime valency must belong to some known families of distance-transitive graphs if the socle of the stabilizer of a vertex is not a 2-transitive linear group with some further restrictions. They posed a problem to determine the connected 2-distance primitive graphs of prime valency when the socle of the stabilizer of a vertex is a 2-transitive linear group. In this paper, we make a crucial progress to this problem by proving that either the graph under consideration in this problem is the folded 5-cube or the socle of the stabilizer of a vertex is PSL(2,q) and the graph has girth 4 and contains no 5-cycles. We also give a short proof of the main result of Jin, Huang and Liu (2019), and we find some new graphs that are not in the list of Jin, Huang and Liu (2019).

New Dimensional Estimates for Subvarieties of Linear Algebraic Groups

New Dimensional Estimates for Subvarieties of Linear Algebraic Groups

Open distal pancreatectomy stump closure by linear stapling versus continuous suturing: a retrospective–prospective comparative study

Background Distal pancreatectomy (DP), performed for removing pancreatic lesions located to the left of the superior mesenteric vein, accounts for ~25% of all pancreatic resections. Most recently reported overall mortality rate of DP is less than 3%. However, albeit technically simpler to perform than pancreaticoduodenectomy, morbidity rate of DP remains substantial. Postoperative pancreatic fistulas (POPFs), which may result in numerous clinically significant and potentially life-threatening complications such as intraabdominal abscess, hemorrhage, and surgical site infection, is the leading cause of morbidity after DP, with an incidence of ~30–60%. Risk factors associated with POPFs after DP include decreased serum albumin levels, concurrent splenectomy, increased body weight, increased duration of surgery, higher American Society of Anesthesiologists score, and impaired renal function. Furthermore, POPFs often translate to significant increases in hospitalization costs. Objective To compare the technique of DP stump closure in two groups of patients; a group underwent stump closure with linear stapling technique and the other group underwent stump closure with continuous suturing technique. Patients and methods This was a retrospective–prospective clinical trial that was conducted at General Surgery Department, Ain Shams University Hospitals on 30 patients who were admitted to specialized hepatobiliary units in Cairo, Egypt for open DP. Between January 2020 and June 2022 with postoperative follow-up period of 2 months. In our study, there were 30 patients (15 in the continuous suturing technique group and 15 in the linear stapling technique group) with a mean age of 45.37±10.662 years. Groups were similar in demographic and clinical characteristics (P>0.05). Results In our study, 13.33% patients in continuous suturing technique group developed clinically relevant POPF (grade B and grade C POPF), whereas 33.33% patients in stapling technique group develop POPF (P=1.00). The amount of the blood loss which was found to be statistically significant as the P value was found less than 0.05 and the t test was found to be −2.396 which mean that the amount of the blood lost was more in the stapler group. There was no statistically significant difference in the rate of clinically relevant POPF (grade B and grade C POPF) with suture closure compared with stapler closure (13.33 vs. 33.33%). There was no statistically significant difference in the risk of intraabdominal abscess. There were three (20%) patients with a biochemical leak in continuous suturing technique group. In stapling technique group, three (20%) patients developed biochemical leaks. There was no significant in difference between groups in terms of biochemical leak. In our study, mortality rates did not differ between stapler and suture closure techniques. Conclusion Our study showed no significant difference between suture and stapler closure of DP stump with respect to POPF or intraabdominal collection after DP. The amount of the blood lost was more in the stapler group.

A conjecture of Mallows and Sloane with the universal denominator of Hilbert series

Abstract A conjecture of Mallows and Sloane conveys the dominance of Hilbert series for finding basic invariants of finite linear groups if the Hilbert series of the invariant ring is of a certain explicit canonical form. However, the conjecture does not hold in general by a well-known counterexample of Stanley. In this article, we give a constraint on lower bounds for the degrees of homogeneous system of parameters of rings of invariants of finite linear groups depending on the universal denominator of Hilbert series defined by Derksen. We consider the conjecture with the universal denominator on abelian groups and provide some criteria guaranteeing the existence of homogeneous system of parameters of certain degrees. In this case, Stanley’s counterexample could be avoided, and the homogeneous system of parameters is optimal.

Effects of foraging mode and group pattern on vigilance behavior in water birds: a case study of mallard and black-winged stilt

Vigilance behavior is affected by many factors including foraging mode and group pattern which we attempted to understand through a case study of mallard Anas platyrhynchos and black-winged stilt Himantopus himantopus in the Hengshui Lake National Nature Reserve, China. Mallard, a swimming bird, forages in irregular and loose circular groups in open water and its foraging mode (with its head underwater looking for food) does not allow for simultaneous monitoring of potential threats above the water surface, which is why its foraging is called “blind period of foraging”. In contrast, black-winged stilt is a wading bird and forages along riversides in long linear foraging groups. Its long beak, neck and legs enable it to scan around while foraging without any obviously blind period. We used the focal sampling method and measured vigilance behavior by the time spent vigilant, vigilance frequency, average duration of vigilance, alert distance and flee distance for the two species. We predicted higher levels of vigilance and longer alert and flee distance in mallard with long blind periods of foraging. Our results indicated significant differences (p

Unit Group of the Group Algebra $\mathbb{F}_qGL(2,7)$

Unit Group of the Group Algebra $\mathbb{F}_qGL(2,7)$

Excitations of bubbling geometries for line defects

The half-Bogomolny-Prasad-Sommerfield (BPS) Wilson line operators in the irreducible representations labeled by the Young diagrams for N=4 U(N) super Yang-Mills theory have gravity dual descriptions. When the number k of boxes of the diagram grows as k∼N2, the bubbling geometries emerge. We evaluate the spectra of quantum fluctuations on the particular bubbling geometry involving the largest degeneracy from the large N and large k limit of the supersymmetric indices decorated by the Wilson lines. The spectra of excitations over multiparticle 1/8- and 1/2-BPS states agree with the numbers of conjugacy classes of general linear group over finite fields while degeneracies of single particle BPS states are given by the general necklace polynomial. The bubbling geometry exhibits a new class of asymptotic degeneracy of states. Published by the American Physical Society 2024

On p -Group Isomorphism: Search-to-Decision, Counting-to-Decision, and Nilpotency Class Reductions via Tensors

In this article, we study some classical complexity-theoretic questions regarding Group Isomorphism ( GpI ). We focus on p -groups (groups of prime power order) with odd p , which are believed to be a bottleneckcase for GpI , and work in the model of matrix groups over finite fields. Our main results are as follows: • Although search-to-decision and counting-to-decision reductions have been known for more than four decades for Graph Isomorphism , they had remained open for GpI , explicitly asked by Arvind and Torán ( EATCS Bull. , 2005). Extending methods from Tensor Isomorphism (TI) (Grochow and Qiao, ITCS 2021), we show moderately exponential-time such reductions within p -groups of class 2 and exponent p . • Despite the widely held belief that p -groups of class 2 and exponent p are the hardest cases of GpI , there was no reduction to these groups from any larger class of groups. Again using methods from TI (ibid.), we show the first such reduction, namely from isomorphismtesting of p -groups of “small” class and exponent p to those of class two and exponent p . For the first results, our main innovation is to develop linear-algebraic analogues of classical graph coloring gadgets, a key technique in studying the structural complexity of Graph Isomorphism . Unlike the graph coloring gadgets, which support restricting to various subgroups of the symmetric group, the problems we study require restricting to various subgroups of the general linear group, which entails significantly different and more complicated gadgets. The analysis of one of our gadgets relies on a classical result from group theory regarding random generation of classical groups (Kantor and Lubotzky, Geom. Dedicata , 1990). For the nilpotency class reduction, we combine a runtime analysis of the Lazard correspondence with TI-completeness results (Grochow and Qiao, ibid.).

Hg4(Te2O5)(SO4): A Giant Birefringent Sulfate Crystal Triggered by a Highly Selective Cation.

Sulfate crystals are often criticized for their low birefringence. The small anisotropic SO4 group is becoming the biggest bottleneck hindering the application of sulfates in optical functional materials. In this study, we report a new method to significantly enhance the birefringence of sulfates. The title compound increases the birefringence recording of sulfates to 0.542@546 nm, which is significantly larger than that of the commercial birefringent crystal of TiO2 (0.306@546.1 nm). At the infrared wavelength, the birefringence of Hg4(Te2O5)(SO4) can be up to 0.400@1064 nm, which is also much larger than the infrared birefringent crystal of YVO4 (0.209@1064 nm). In addition, it also has a wide transparency range, high thermal stability, and excellent environmental stability, making it a potential birefringent material. Hg4(Te2O5)(SO4) features a novel two-dimensional layered structure composed of [Hg4(Te2O5)]2+ layers separated by isolated (SO4)2- tetrahedra. This compound was designed by introducing a highly selective cation in a tellurite sulfate system. The low valence low coordination cations connect with tellurite groups only, making the sulfate isolated in the structure. The steric repulsive action of the isolated SO4 tetrahedra may regulate the linear and lone pair groups arranged in a way that favors large birefringence. This method can be proven by theoretical calculations. PAWED studies showed that the large birefringence originated from the synergistic effect of (Hg2O2)2-, (Te2O5)2-, and (SO4)2- units, with a contribution ratio of 42.17, 37.92, and 19.88%, respectively. Our work breaks the limitation of low birefringence in sulfates and opens up new possibilities for their application as birefringent crystals.

A comonadicity theorem for partial comodules

We show that the category of partial comodules over a Hopf algebra H is comonadic over Vectk and provide an explicit construction of this comonad using topological vector spaces. The case when H is finite-dimensional is treated in detail. A study of partial representations of linear algebraic groups is initiated; we show that a connected linear algebraic group does not admit partiality.

On abelian unitary involutions of crossed products

In the theory of classical linear algebraic groups, of importance are special unitary groups of non-commuted involution crossed products with division. The description of these groups largely depends on the involution type of these products. The class of Abelian involution crossed products is considered and the criterion for their existence is set provided that unitary bases (with respect to these involutions) are present in these products.

Spectral equivalence of smooth group schemes over principal ideal local rings

Let G be a smooth linear group scheme of finite type. For any positive integer k and a finite field F, let Wk(F) be the ring of Witt vectors of length k over F. We show that the group algebras of G(F[t]/(tk)) and G(Wk(F)) are isomorphic (i.e. the multi-sets of the dimensions of the irreducible representations are equal) for any positive integer k and finite field F with large enough characteristic. We also prove that if charF is large enough, then the cardinality of the set {dim⁡ρ|ρ∈irr(G(F))} is bounded uniformly in F.

Special Unipotent Representations of Simple Linear Lie Groups of Type A

Special Unipotent Representations of Simple Linear Lie Groups of Type A

St. Petersburg School of Linear Groups: II. Early Works by Suslin

St. Petersburg School of Linear Groups: II. Early Works by Suslin

Graphs of schemes associated to group actions

Let $X$ be a proper algebraic scheme over an algebraically closed field. We assume that a torus $T$ acts on $X$ such that the action has isolated fixed points. The $T$-graph of $X$ can be defined using the fixed points and the one-dimensional orbits of the $T$-action. If the upper Borel subgroup of the general linear group with maximal torus $T$ acts on $X$, then we can define a second graph associated to $X$, called the $A$-graph of $X$. We prove that the $A$-graph of $X$ is connected if and only if $X$ is connected. We use this result to give proof of Hartshorne's theorem on the connectedness of the Hilbert scheme in the case of $d$ points in $\mathbb{P}^{n}$. .

A vanishing theorem for varieties with finitely many solvable group orbits

AbstractWe reprove and generalize a result proved by Yavin that the intersection cohomology groups of a toric variety with coefficient in a nontrivial rank one local system vanish. We prove a similar vanishing result for a certain class of varieties on which a connected linear solvable group acts, including all spherical varieties.

Strong Gelfand pairs of SL(2, pn )

A strong Gelfand pair (G, H) is a group G together with a subgroup H such that every irreducible character of H induces a multiplicity-free character of G. We classify the strong Gelfand pairs of the special linear groups SL ( 2 , p n ) where p is a prime.

Lie invariant Frobenius lifts

We begin with the observation that the p-adic completion of any affine elliptic curve with ordinary reduction possesses Frobenius lifts ϕ that are “Lie invariant mod p” in the sense that the “normalized” action of ϕ on 1-forms preserves mod p the space of invariant 1-forms. Our main result is that, after removing the 2-torsion sections, the above situation can be “infinitesimally deformed” in the sense that the above mod p result has a mod p2 analogue. We end by showing that, in contrast with the case of elliptic curves, the following holds: if G is a linear algebraic group over a number field and if G is not a torus then for all but finitely many primes p the p-adic completion of G does not possess a Frobenius lift that is Lie invariant mod p.

The relational complexity of linear groups acting on subspaces

Abstract The relational complexity of a subgroup 𝐺 of Sym ⁢ ( Ω ) \mathrm{Sym}({\Omega}) is a measure of the way in which the orbits of 𝐺 on Ω k \Omega^{k} for various 𝑘 determine the original action of 𝐺. Very few precise values of relational complexity are known. This paper determines the exact relational complexity of all groups lying between PSL n ⁢ ( F ) \mathrm{PSL}_{n}(\mathbb{F}) and PGL n ⁢ ( F ) \mathrm{PGL}_{n}(\mathbb{F}) , for an arbitrary field 𝔽, acting on the set of 1-dimensional subspaces of F n \mathbb{F}^{n} . We also bound the relational complexity of all groups lying between PSL n ⁢ ( q ) \mathrm{PSL}_{n}(q) and P ⁢ Γ ⁢ L n ⁢ ( q ) \mathrm{P}\Gamma\mathrm{L}_{n}(q) , and generalise these results to the action on 𝑚-spaces for m ≥ 1 m\geq 1 .

An inductive approach to representations of general linear groups over compact discrete valuation rings

In his seminal Lecture Notes in Mathematics published in 1981, Andrey Zelevinsky introduced a new family of Hopf algebras which he called PSH-algebras. These algebras were designed to capture the representation theory of the symmetric groups and of classical groups over finite fields. The gist of this construction is to translate representation-theoretic operations such as induction and restriction and their parabolic variants to algebra and coalgebra operations such as multiplication and comultiplication. The Mackey formula, for example, is then reincarnated as the Hopf axiom on the algebra side. In this paper we take substantial steps to adapt these ideas for general linear groups over compact discrete valuation rings. We construct an analogous bialgebra that contains a large PSH-algebra that extends Zelevinsky's algebra for the case of general linear groups over finite fields. We prove several base change results relating algebras over extensions of discrete valuation rings.