Academic literature on the topic 'Mathematics (probability theory, representation theory of finite groups, and algebraic combinatorics)'

Understanding the algebraic and geometric structures that form in diverse mathematical areas depends heavily on the study of representation theory. The significance and uses of representation theory in both algebra and geometry are briefly discussed in this abstract.The primary goal of representation theory is to understand how linear transformations on vector spaces can represent abstract algebraic objects like groups, rings, and algebras. Representation theory offers a strong framework to analyse and interact with these structures using the methods and tools of linear algebra by linking algebraic structures with linear transformations.The representation theory has significant effects on algebra. Through the examination of the representations that go along with a group, it allows us to examine its composition and behaviour. One can learn more about the internal structures and underlying symmetries of groups by breaking representations down into irreducible parts. This has implications for number theory, combinatorics, and quantum physics, among other fields.Understanding symmetries and transformations of geometric objects in geometry depends critically on representation theory. The investigation of shape and space symmetry is made possible by representation theory, which links geometric objects with linear transformations. This has uses in a variety of disciplines, including physics' study of symmetry groups, differential geometry, and crystallography.A crucial area of research, representation theory has extensive uses in both algebra and geometry. It is indispensable in many branches of mathematics and opens up new directions for research and discovery because it can reveal the underlying structures and symmetries of abstract algebraic objects.

Abstract The Alperin–McKay conjecture is a longstanding open conjecture in the representation theory of finite groups. Späth showed that the Alperin–McKay conjecture holds if the so-called inductive Alperin–McKay (iAM) condition holds for all finite simple groups. In a previous paper, the author has proved that it is enough to verify the inductive condition for quasi-isolated blocks of groups of Lie type. In this paper, we show that the verification of the iAM-condition can be further reduced in many cases to isolated blocks. As a consequence of this, we obtain a proof of the Alperin–McKay conjecture for $2$ -blocks of finite groups with abelian defect.

Abstract Orthogonality is a fundamental theme in representation theory and Fourier analysis. An orthogonality relation for characters of finite abelian groups (now recognized as an orthogonality relation on $\mathrm {GL}(1)$ ) was used by Dirichlet to prove infinitely many primes in arithmetic progressions. Orthogonality relations for $\mathrm {GL}(2)$ and $\mathrm {GL}(3)$ have been worked on by many researchers with a broad range of applications to number theory. We present here, for the first time, very explicit orthogonality relations for the real group $\mathrm {GL}(4, \mathbb R)$ with a power savings error term. The proof requires novel techniques in the computation of the geometric side of the Kuznetsov trace formula.

The aim of this paper is to develop a theory of finite transformation monoids and in particular to study primitive transformation monoids. We introduce the notion of orbitals and orbital digraphs for transformation monoids and prove a monoid version of D. Higman's celebrated theorem characterizing primitivity in terms of connectedness of orbital digraphs. A thorough study of the module (or representation) associated to a transformation monoid is initiated. In particular, we compute the projective cover of the transformation module over a field of characteristic zero in the case of a transitive transformation or partial transformation monoid. Applications of probability theory and Markov chains to transformation monoids are also considered and an ergodic theorem is proved in this context. In particular, we obtain a generalization of a lemma of P. Neumann, from the theory of synchronizing groups, concerning the partition associated to a transformation of minimal rank.

In recent years it has been noted that a number of combinatorial structures such as real and complex hyperplane arrangements, interval greedoids, matroids and oriented matroids have the structure of a finite monoid called a left regular band. Random walks on the monoid model a number of interesting Markov chains such as the Tsetlin library and riffle shuffle. The representation theory of left regular bands then comes into play and has had a major influence on both the combinatorics and the probability theory associated to such structures. In a recent paper, the authors established a close connection between algebraic and combinatorial invariants of a left regular band by showing that certain homological invariants of the algebra of a left regular band coincide with the cohomology of order complexes of posets naturally associated to the left regular band. The purpose of the present monograph is to further develop and deepen the connection between left regular bands and poset topology. This allows us to compute finite projective resolutions of all simple modules of unital left regular band algebras over fields and much more. In the process, we are led to define the class of CW left regular bands as the class of left regular bands whose associated posets are the face posets of regular CW complexes. Most of the examples that have arisen in the literature belong to this class. A new and important class of examples is a left regular band structure on the face poset of a CAT(0) cube complex. Also, the recently introduced notion of a COM (complex of oriented matroids or conditional oriented matroid) fits nicely into our setting and includes CAT(0) cube complexes and certain more general CAT(0) zonotopal complexes. A fairly complete picture of the representation theory for CW left regular bands is obtained.

This is the first of a series of papers concerning what might be thought of as ‘locally grouped spaces’, in a loose analogy with the locally ringed spaces of algebraic geometry. The spaces that we have in mind are simplicial sets that generalise the simplicial sets that underlie and determine the classifying spaces of finite (or compact) groups. If the analogy is pursued, then the role of ‘structure sheaf’ is provided by the ‘fusion systems’ associated with these spaces. Our approach here will be purely algebraic and combinatorial, so we will not be concerned with topological realisations. All of the groups to be considered will be finite; but a parallel series of papers representing some joint work with Alex Gonzalez will considerably broaden the scope.

Abstract We study reductive subgroups H of a reductive linear algebraic group G – possibly nonconnected – such that H contains a regular unipotent element of G. We show that under suitable hypotheses, such subgroups are G-irreducible in the sense of Serre. This generalises results of Malle, Testerman and Zalesski. We obtain analogous results for Lie algebras and for finite groups of Lie type. Our proofs are short, conceptual and uniform.

Abstract We give a construction that takes a simple linear algebraic group G over a field and produces a commutative, unital, and simple non-associative algebra A over that field. Two attractions of this construction are that (1) when G has type $E_8$ , the algebra A is obtained by adjoining a unit to the 3875-dimensional representation; and (2) it is effective, in that the product operation on A can be implemented on a computer. A description of the algebra in the $E_8$ case has been requested for some time, and interest has been increased by the recent proof that $E_8$ is the full automorphism group of that algebra. The algebras obtained by our construction have an unusual Peirce spectrum.

Abstract We provide a microlocal necessary condition for distinction of admissible representations of real reductive groups in the context of spherical pairs. Let ${\mathbf {G}}$ be a complex algebraic reductive group and ${\mathbf {H}}\subset {\mathbf {G}}$ be a spherical algebraic subgroup. Let ${\mathfrak {g}},{\mathfrak {h}}$ denote the Lie algebras of ${\mathbf {G}}$ and ${\mathbf {H}}$ , and let ${\mathfrak {h}}^{\bot }$ denote the orthogonal complement to ${\mathfrak {h}}$ in ${\mathfrak {g}}^*$ . A ${\mathfrak {g}}$ -module is called ${\mathfrak {h}}$ -distinguished if it admits a nonzero ${\mathfrak {h}}$ -invariant functional. We show that the maximal ${\mathbf {G}}$ -orbit in the annihilator variety of any irreducible ${\mathfrak {h}}$ -distinguished ${\mathfrak {g}}$ -module intersects ${\mathfrak {h}}^{\bot }$ . This generalises a result of Vogan [Vog91]. We apply this to Casselman–Wallach representations of real reductive groups to obtain information on branching problems, translation functors and Jacquet modules. Further, we prove in many cases that – as suggested by [Pra19, Question 1] – when H is a symmetric subgroup of a real reductive group G, the existence of a tempered H-distinguished representation of G implies the existence of a generic H-distinguished representation of G. Many of the models studied in the theory of automorphic forms involve an additive character on the unipotent radical of the subgroup $\bf H$ , and we have devised a twisted version of our theorem that yields necessary conditions for the existence of those mixed models. Our method of proof here is inspired by the theory of modules over W-algebras. As an application of our theorem we derive necessary conditions for the existence of Rankin–Selberg, Bessel, Klyachko and Shalika models. Our results are compatible with the recent Gan–Gross–Prasad conjectures for nongeneric representations [GGP20]. Finally, we provide more general results that ease the sphericity assumption on the subgroups, and apply them to local theta correspondence in type II and to degenerate Whittaker models.

Abstract We define a notion of height for rational points with respect to a vector bundle on a proper algebraic stack with finite diagonal over a global field, which generalizes the usual notion for rational points on projective varieties. We explain how to compute this height for various stacks of interest (for instance: classifying stacks of finite groups, symmetric products of varieties, moduli stacks of abelian varieties, weighted projective spaces). In many cases, our uniform definition reproduces ways already in use for measuring the complexity of rational points, while in others it is something new. Finally, we formulate a conjecture about the number of rational points of bounded height (in our sense) on a stack $\mathcal {X}$ , which specializes to the Batyrev–Manin conjecture when $\mathcal {X}$ is a scheme and to Malle’s conjecture when $\mathcal {X}$ is the classifying stack of a finite group.

This thesis studies mixing times for three random walk models. Specifically, these are random walks on the alternating group, the group of signed permutations and the complete monomial group. The details for the models are given below: The random walk on the alternating group: We investigate the properties of a random walk on the alternating group $A_n$ generated by $3$-cycles of the form $(i, n − 1, n)$ and $(i, n, n − 1)$. We call this the transpose top-2 with random shuffle. We find the spectrum of the transition matrix of this shuffle. We obtain the sharp mixing time by proving the total variation cutoff phenomenon at $(n − 3/2) \log n$ for this shuffle. The random walk on the group of signed permutations: We consider a random walk on the hyperoctahedral group $B_n$ generated by the signed permutations of the form $(i, n)$ and $(−i, n)$ for $1 ≤ i ≤ n$. We call this the flip-transpose top with random shuffle on $B_n$. We find the spectrum of the transition probability matrix for this shuffle. We prove that this shuffle exhibits the total variation cutoff phenomenon with cutoff time $n \log n$. Furthermore, we show that a similar random walk on the demihyperoctahedral group $D_n$ generated by the identity signed permutation and the signed permutations of the form $(i, n)$ and $(−i, n)$ for $1 ≤ i < n$ also has a cutoff at $(n − 1/2) \log n$. The random walk on the complete monomial group: Let $G_1 ⊆ · · · ⊆ G_n ⊆ · · ·$ be a sequence of finite groups with $|G_1| > 2$. We study the properties of a random walk on the complete monomial group $G_n\wrS_n$ generated by the elements of the form $(e, . . . , e, g; id)$ and $(e, . . . , e, g^{−1} , e, . . . , e, g; (i, n))$ for $g ∈ G_n, 1 ≤ i < n$. We call this the warp-transpose top with random shuffle on $G_n\wr S_n$. We find the spectrum of the transition probability matrix for this shuffle. We prove that the mixing time for this shuffle is of order $n \log n + (1/2) n \log(|G_n | − 1)$. We also show that this shuffle satisfies cutoff phenomenon with cutoff time $n \log n$ if $|G_n| = o(n^{δ})$ for all $δ > 0$.