ICTS

Abstracts for GTACM16 Discussion meeting (Nov. 11-14, 2016)

• Valeriy Bardakov
  Title: Linearity problem for non-abelian tensor product
  Abstract: A group G Is called linear if there is an embedding of G into general linear group GL(n, k) over some field k. I formulated some results on the linearity of some non-abelan tensor products and, in particular, for some tensor squares. Also, I construct concrete representations for tensor squares of some groups.
  
  This is a joint work with M. Neshchadim.
  
   
• Silvio Dolfi
  Title: On the character degree graph of finite groups
  Abstract: I will discuss some results relating the structure of finite groups to their character degree graph. In particular, I will present a description of the solvable groups with character degree graph of diameter three and discuss the existence of large complete subgraphs in the character degree graph of solvable groups.
  
   
• Heiko Dietrich
  Title: Group embeddings of partial Latin squares
  Abstract: A partial Latin square (PLS) is a matrix in which some cells may be empty and in which each filled cell contains one symbol from an underlying alphabet; symbols must be arranged so that no symbol occurs more than once within any row or column. A PLS P can be embedded in a group G if, informally, a copy of P can be found within the Cayley table of G. We discuss a few results on embeddability and, in particular, answer a question of Hirsch and Jackson (2012) who asked for the smallest PLS which can be embedded in an infinite group, but in no finite group. This is based on work of / joint work with Bridget Webb and Ian Wanless.
  
   
• Graham Ellis
  Title: Group theoretic structures for van Kampen theorems
  Abstract:
  
   
• Gerhard Hiss
  Title: Imprimitive irreducible representations of finite quasisimple groups
  Abstract: This is a survey on a joint research project with William J. Husen and Kay Magaard. The aim is to classify the absolutely irreducible, imprimitive representations of all finite quasisimple groups. 
  I will report on the motivation behind this project and on the recent completion of this task for representations in characteristic 0.
  
   
• Alexander Hulpke
  Title: Calculations with Matrix groups over the integers
  Abstract: For matrix groups over the integers, reduction by a modulus m is a fundamental algorithmic tool. I will investigate how it can be used to study such groups on the computer, to test finiteness or finite index. Particular emphasis is given to Arithmetic groups, that is subgroups of SL_n(Z) or Sp_n(Z) of finite index. For determining such an index the structure of classical groups over residue class rings Z/mZ, and the representation theory of classical groups become the major tools. This is joint work with A. Detinko and D. Flannery (both NUI Galway).
  
   
• Alexander Ivanov
  Title: Majorana representations of the symmetric and alternating groups
  Abstract: The Monster group M is the largest sporadic simple group. Its minimal complex representation has dimension 196883 and M preserves a non-associative commutative algebra product on the underlying vector space, known as the Norton algebra. It has been observes by J. McKay in late 1970s that this dimension is linear term of the q-expansion of the modular invariant J, minus 1. A Monster-invariant algebra on a 196884-dimensional space which restricts to the Norton algebra was analysed by J. Conway and used by R. Griess in his construction of the Monster. Thus we talk about 196884-dimensional Conway GriessNorton algebra for the Monster. The current state of the Monstrous Moonshine is that there is an in- finite dimensional algebra V # which belongs to the class of so-called Vertex Operator Algebras of which M is the automorphism group. The dimensions of the graded pieces of V # are the coefficients of J. The Vertex Operator axiomatic involves infinitely many products on the underlying vector space and restricting one of then to the first nontrivial graded piece is the ConwayGriessNorton algebra. There is an important notion of conformal vector in a Vertex Operator Algebras and the conformal vectors with central charge 1/2 correspond elements of order 2 in M from one of two conjugacy classes, centralized by the central extension of the Baby Monster, which is the second largest sporadic simple group. M. Miyamoto have shown that an involutory automorphism can be associated with a conformal vector of central charge 1/2 subject to certain constrains on the algebra, fulfilled by V #. S. Sakuma have classified the subalgebras generated by two conformal vectors of central charge 1/2. The origination of the Majorana Theory is the observation that Miyamoto involutions and Sakuma classification can be achieved already in the finite dimensional algebra for the Monster and its provides an axiomatic approach studying of subgroup structure of the Monster. The principal current goal of the Majorana Theory is to classify the subalgebas of the Monster algebra generated by the Majorana axis of the symmetric and alternating subgroups of the Monster.
   

• Radha Kessar
  Title: Rationality and Morita equivalence for blocks of finite groups
  Abstract:​ Let k be a field and let σ be an automorphism of k. For a finite dimensional k-algebra A the σ -twist ^σA is the k-algebra which equals A as ring, but where scalar multiplication is given by λ.x = σ ⁻¹ (λ)x. Then A and ^σA are not Morita equivalent in general. On the other hand, since A and ^σA are isomorphic as rings, they have the same numerical invariants and hence it is not easy to tell them apart. In my lectures, I will discuss this theme for blocks of finite groups, and its relevance for some of the famous global-local conjectures in modular representation theory.
  
   
• Elena Konstantinova
  Title: Hamiltonicity of Cayley graphs and Gray codes: open problems
  Abstract: In 1970 L´asl´o Lov´asz [1] posed the following question: Does every connected vertex-transitive graph with more than two vertices have a Hamiltonian path (a path that visits each vertex exactly once in a graph)? There are only four vertex-transitive graphs on more than two vertices which do not have a Hamiltonian cycle, and all of these graphs have a Hamiltonian path. Furthermore, it was noted that no one of these graphs is a Cayley graph. So, there is one more conjecture: Every connected Cayley graph on a finite group has a Hamiltonian cycle. In this talk we consider hamiltonicity of Cayley graphs as well as their generalized Gray codes. It is known that there is a connection between hamiltonicity of graphs and combinatorial Gray codes [2]. We discuss open problems in this area. In particular, we consider the Pancake graph on the symmetric group which is generated by prefix-reversals and some of its generalizations. The hamiltonicity of the Pancake graph was known since 1984 when S. Zaks has introduced the algorithm of successive generation of permutations by suffix–reversals which are obviously isomorphic to prefix–reversals [3]. In 2013 A. Williams and J. Sawada suggested to consider a new class of greedy Prefix–reversal Gray codes using greedy sequences [4]. However, the question of existence of other greedy sequences which produce Hamiltonian cycle is unresolved [5]. One more interesting question concerning the existence of Gray codes arises for the Big−3 Pancake graph [6] which is generated by three biggest prefix-reversals. The following conjecture was formulated recently by J. Sawada and A. Williams: there exist cyclic Gray codes in the Big − 3 Pancake graph [7].

   

  References

1. L. Lov´asz, Problem 11 in: Combinatorial structures and their applications, (Proc. Calgary Intern. Conf., Calgary, Alberta, 1969 ), Gordon and Breach, New York, 1970, pp.243–246.
2. C. Savage, A survey of combinatorial Gray codes, SIAM Review 39 (1996) 605–629.
3. S. Zaks, A new algorithm for generation of permutations, BIT 24 (1984) 196–204.
4.  A. Williams, J. Sawada, Greedy Pancake Flipping, Electronic Notes in Discrete Mathematics 44 (2013) 357–362.
5. E. Konstantinova, A. Medvedev, Independent even cycles in the Pancake graph and greedy Prefix-reversal Gray codes, Graphs and Combinatorics 32 (2016) 1965–1978.
6. D. Bass, I. Sudborough, Pancake problems with restricted prefix reversal and some corresponding Cayley networks, J. Parallel Distrib. Comput. 63(3) (2003) 327–336.
7. J. Sawada, A. Williams, Successor rules for flipping pancakes and burnt pancakes, Theor. Comp. Science 609 (2016) 60–75.
    

• Ravi Kulkarni
  Title: Algorithmic Construction of Representations of Finite Solvable Groups Over Arbitrary Fields.
  Abstract: Let G be a finite group and F a field of characteristic p which is prime to |G|. The usual representation theory in text books is developed when F is algebraically closed. We shall briefly review the modifications needed when F is not algebraically closed. In the general case, the arithmetic of F plays an important role. We are interested in actually constructing the matrices for representations for a generating system for G. From this perspective, even the case of abelian groups shows interesting phenomena. When G is solvable, there exists a useful system of generators, called a ”long system of generators”. In terms of this system of generators we shall give a systematic procedure for constructing matrices for generators, and the primitive central idempotents of the group algebra F[G]. There are elementary, interesting, connections with algebra, number theory and algebraic geometry.
   

• Patrizia Longobardi
  Title: On Sums of Element Orders in Finite Groups
  Abstract: Let G be a periodic group. The problem of obtaining information about the structure of G by looking at the orders of its elements has been considered by many authors, from many different points of view. In this talk we consider a finite group G, and we study the function on the element orders of G defined by
   
  
  ψ(G) =  Σ_x∈G o(x),
  
  where o(x) denotes the order of the element x.
  In 2009 H. Amiri, S.M. Jafarian Amiri and M. Isaacs proved that if G has order n and Cn denotes the cyclic group of order n, then 
  
  
  ψ(G) ≤ ψ(C_n), 
  
  and
  
  ψ(G) = ψ(C_n) if and only if G≃C_n. 
  
  
  Other results have been obtained by H. Amiri, S.M. Jafarian Amiri, M. Amiri, Y. Marefat, A. Iranmanesh, A. Tehranian, R. Shen, G. Chen and C. Wu. 
  
  I will discuss some new results concerning the function ψ, jointly obtained with Marcel Herzog and Mercede Maj. In particular I will present some better upper bounds for ψ(G) when G is not cyclic. 
  
  Some other functions on the orders of the elements of a finite group G have been recently investigated by M. Garonzi and M. Patassini.
   

• Primoz Moravec
  Title: Commutativity Preserving Extensions of Groups
  Abstract: In parallel to the classical theory of central extensions of groups, we develop a version for extensions that preserve commutativity. It is shown that the Bogomolov multiplier is a universal object parametrizing such extensions of a given group. Maximal and minimal extensions are inspected, and a connection with commuting probability is explored. Such considerations produce bounds for the exponent and rank of the Bogomolov multiplier. This is joint work with Urban Jezernik.
   

• Gabriel Navarro
  Title: New Local Properties in the Character Table
  Abstract: We discuss several local properties of finite groups that can be detected in the character table.
   

• I. B. S. Passi
  Title: Group Rings and Jordan Decomposition
  Abstract: I will give a survey of the Jordan decomposition problem in integral group rings.
   

• Dipendra Prasad
  Title: Distinguished representations for classical groups
  Abstract: A representation of a group G is said to be distinguished by a subgroup H if it has a fixed vector under H. Study of distinguished rep’s is of importance in finite groups, Harmonic analysis on real and p-adic groups, and number theory. This talk will focus on classical groups over finite fields.
   

• Paramesh Sankaran
  Title: The BNS invariant and applications
  Abstract: To a finitely generated group, Bieri, Neumann, and Strebel associated an open subset of a sphere of dimension n − 1 where n is the rank of the abelianization of the given group. We will review the definition of the Bieri-Neumann-Strebel invariant and its generalizations. We shall illustrate its application to the twisted conjugacy problem for certain groups of P L-homeomorphisms of the interval.
   

• Carlo M. Scoppola
  Title: More on p-groups of small breadth
  Abstract: This is joint work with Andrea Cupaiolo, Norberto Gavioli, and Alessandro Morresi Zuccari. Revisiting previous work of ours, of Parmeggiani-Stellmacher, and of other authors, we were able to compute exactly some isoclinic invariants for classes of p-groups whose conjugacy classes have length at most p 3 , where p is an odd prime.
   

• Anupam K. Singh
  Title: Conjugacy classes of centralizers in algebraic groups
  Abstract: Let G be a semisimple algebraic group defined over a field k. Denote the k points of G by G(k). Two elements of G(k) are said to be z-equivalent if their centralizers are conjugate in G(k). Steinberg proved that when k is an algebraically closed field of good characteristic, a semisimple algebraic group has finitely many z-classes. We ask the question that for what fields the finiteness of zclasses is still true for simple algebraic groups. We present the result that over fields k with property that it has only finitely many extensions of any finite degree, the classical groups have finitely many z-classes. Counting the number of z-classes is relevant in Representation theory of finite groups of Lie type and in Geometry. This work is in collaboration with my student Bhunia and is extension of previous work done by Gongopadhyay and Kulkarni.
   

• Pooja Singla
  Title: On characterization of monomial irreducible representations of discrete supersolvable groups
  Abstract:  A representation of a group is called monomial if it is induced from a one dimensional representation. A group is called monomial if its every irreducible representation is monomial. It is well known that every finite nilpotent group is monomial. From the orbit method of Kirillov, it also follows that every finitely generated nilpotent Lie group is monomial. However I.D.Brown, while considering unitary representations, proved that finitely generated discrete nilpotent groups are not necessarily monomial. He further characterized those unitary irreducible representations of finitely generated discrete nilpotent groups that are monomial. A conjecture of A.N.Parshin states that Brown's characterization extends to arbitrary (not necessarily unitary) irreducible representations of finitely generated discrete nilpotent groups. This conjecture was recently proved by Beloshapka-Gorchinskiy. In this talk we will describe these results. In the end, we will show that these results extend to discrete supersolvable groups. This is based on joint work with E.K.Narayanan.
   

• Maneesh Thakur
  Title: Retract rationality of some (exceptional) group varieties
  Abstract: Let F be a field having at least four elements. The group SL(n, F) of n by n matrices of determinant 1 with F-entries, is generated by unipotent matrices (i.e. those which have all eigenvalues equal to 1) and is simple modulo its center. Let G be a simple and simply connected group over F (e.g. G = SL(n)). Kneser-Tits conjecture predicts that if G is isotropic over F ( equivalently, if G(F) contains non-trivial unipotents ), then G(F) is simple modulo its center. The conjecture is well known to be false. The obstruction to the validity of this conjecture lies in the Whitehead group of G. One is interested in computing this obstruction. If this obstruction is trivial, then G(F) is simple modulo its center, providing interesting examples of (abstract) simple groups. We will discuss two geometric notions for algebraic groups, called R-equivalence (due to Manin), and retract rationality and mention a link with Whitehead groups, and finally report on some results we have obtained for some E-type exceptional groups.