Technion, Department of Mathematics
Algebra
seminar
Spring
semester 2011  The seminar will meet on Thursdays at 13:30 in Amado 619.
DATE 
SPEAKER 
June
23 
Daniel Rabayev Israel Math
Union Meeting Bar IlanNo Seminar Yossi Cohen (Hebrew), Jack Sonn
Note special date time and room 
June 9 
YOM GESHER – No seminar 
May 19 
No
seminar 
DEPARTMENT OF
MATHEMATICS
TECHNION
==================================
Special Number Theory Seminar
Two Talks
Note special date, time and room
==================================
1.
SPEAKER:
Yossi Cohen, Technion
TITLE: On the sequence gcd(a^n+1, b^n+1) (In
Hebrew).
DATE: Monday, June 13, 2011
PLACE: Room 619
Amado Building, Technion.
TIME: 12:3013:20
2.
SPEAKER:
Jack Sonn, Technion
TITLE: On the sequence gcd(Phi_N(a^n), Phi_N(b^n));
an
application
of the Chebotarev density theorem
DATE: Monday, June 13, 2011
PLACE: Room
Amado Building, Technion.
TIME: 13:3014:20
ABSTRACT:

 
Speaker: Robert Shwartz, Bar Ilan
University.
Title: "Interesting properties of the
Atype Affine Coxeter
groups".

    

SPEAKER: Darrel Haile,
Indiana University and Technion.
TITLE: Noninvertible Galois cohomology
DATE: Thursday,
May 12 2011
PLACE: Room
719 Amado Building, Technion.
TIME: 13:30
TIME: 13:30
ABSTRACT
In this lecture I want to introduce noninvertible Galois cohomology.
As a motivating problem consider the following
situation. Let K/F be a
finite separable extension of fields and let A be
a central simple
Kalgebra. We are interested
in Falgebras B that contain A and are
"tightly connected" to A. I
will give a formal definition in the
lecture but an example would be a central simple
Falgebra B containing
A such that A is the full centralizer of K in B.
Such a central simple
Falgebra B may not exist. The
classification of the Falgebras B that
do exist, for a fixed A,
is an application of the noninvertible Galois
cohomology I will discuss.
      
TECHNION
FACULTY OF MATHEMATICS
ALGEBRA SEMINAR
SPEAKER: Avi Goren, Technion.
TITLE:
Upper bound for the relative part of stabilizers in conjugacy classes of finite permutation
groups.
DATE: Thursday, May. 5, 2011
TIME: 13:30
PLACE: Amado 719
Abstract:
Let G be a finite transitive permutation group on a finite set S. Let A be a nonempty subset of S
and denote the pointwise stabilizer of A in G by C(A). Let T be a normal subset of G not
containing 1. Let U be a Ginvariant subset o f S.
Well demonstrate a connection between the relative part of A in U and the relative part of C(A) in
T.
Same connection holds if G is a simple group .
 
SPEAKER:
Shlomi Gelaki,
Technion
TITLE: Virtually indecomposable tensor categories
DATE: Thursday, April 28th 2011
PLACE: Room
719 Amado Building, Technion.
TIME: 13:30
ABSTRACT:
JP. Serre proved the following: 1) For any group G, the
spectrum of the Grothendieck ring of its
representation category Rep(G) over any field k, is
connected. 2) The same is true when G is replaced by any Lie algebra and k has
characteristic 0.
JP. Serre asked whether this remains true for any Tannakian category, and in
particular for Lie algebras over k with positive characteristic (a case of
particular interest to him).
Inspired
by the this, we say that a tensor category over k is
virtually indecomposable if its Grothendieck ring
contains no nontrivial central idempotents. In my
talk I will describe our following results: tensor categories with the Chevalley property, representation categories of affine
(super)group schemes and formal (super)groups, and symmetric tensor categories
of exponential growth in characteristic 0, are all virtually indecomposable
tensor categories.
As
special cases of these results, we obtain an alternative proof to Serre's theorem and a positive answer to his questions.

SPEAKER:
Danny Neftin
TITLE: Admissibility of finite groups over number
fields
DATE: Thursday, April 14 2011
PLACE: Room
719 Amado Building, Technion.
TIME: 13:30
ABSTRACT:
A finite
group G is called Kadmissible if there is a Galois Gextension L/K and a
division algebra D with center K for which L is a maximal subfield of D (in
which case, D is called a Gcrossed product division
algebra).
Over
number fields K, we present necessary conditions for Kadmissibility, called Kpreadmissibility, that are easier to verify. We shall show
that in many cases preadmissibility implies
admissibility, discuss the case of abelian groups A
in detail and use tools from class field theory to determine the special cases
in which A is Kpreadmissibile but not Kadmissible.
Finally, we shall use our knowledge on preadmissibility
to study an arithmetic equivalence relation induced by admissibility.

SPEAKER:
Crystal Hoyt, BarIlan
University
TITLE: Finite Walgebras and primitive ideals
DATE: Thursday, April 7 2011
PLACE: Room
719 Amado Building, Technion.
TIME: 13:30
I will
begin with a definition of finite Walgebras and then give a partial survey of
the literature. A finite Walgebra U(g,e) is a certain associative algebra which can be
constructed from the universal enveloping algebra U(g) of a semisimple
Lie algebra g and nilpotent element e of
g. These algebras first arose from the
work of Kostant on Whittaker modules in 1978. It was shown in 2002 by Premet
and GanGinzburg that U(g,e) can be realized as a quantization of theSlodowy slice to the nilpotent Ad G orbit O_e.
I will discuss the connection of finite
Walgebras to primitive ideals of U(g) and Goldie rank
polynomials, and in particular, the recent work of Premet,
Losev and Brundan.

SPEAKER: Doron Puder, Hebrew University
TITLE: On Primitive Free Words: A New Algorithm to
Detect them and Measure Preservation.
DATE: Thursday, March 31, 2011
PLACE: Room 719 Amado Building, Technion.
TIME: 13:30
ABSTRACT:
An element $w \in F_k$, the free group on $k$
generators, is called primitive* if it belongs to a basis (i.e. a minimal
generating set) of $F_k$. In the first part of the
talk I will present briefly a new elementary algorithm to determine whether a
given word is primitive.
In the
second part, the results from the first one will be used to show a tight
connection between primitivity and measurepreservation. A free word $w \in F_k$
is called *measure preserving* if for every finite group $G$, the word map $w: G^k \to G$ induces uniform distribution on $G$ (given
uniform distribution on $G^k$).
It is an
easy observation that a primitive word is measure preserving. It is conjectured
that the converse is also true. We prove it for $F_2$. In fact, both notions (primitivity, measure preservation) can be naturally
extended to sets of free words. The strategy weuse
actually shows that for sets of size at least $k1$ in $F_k$,
the two notions indeed coincide.

·
Thursday, March 24, 13:30. Room:
Amado 719.
Luda Marcus, Technion
Word Problem for Inverse Monoids Presented
by a Single Relator.
This is joint work with Mark Berman, Uri Onn, and Pirita Paajanen.
Since Magnus it has been well known that onerelator groups have a
decidable word problem. However, solvability of the word problem in onerelator
monoids is far from being completely studied. Only
few examples of inverse monoids with solvable word
problem are known. Recently, the solvability of the word problem in inverse monoids with a single sparse relator has been announced by Hermiller, Lindblad and Meakin.
We consider certain onerelator inverse monoids.
In our attempt to solve the word problem, we relay on the result of Ivanov, Margolis and Meakin which
states that the word problem for the inverse $E$unitary onerelator monoid is decidable if the membership problem for the
prefix monoid is decidable. Thus, we first solve the
membership problem for the prefix monoid and then
apply the theorem to solve the word problem. Our methods employ van Kampen diagrams and word combinatorics.Thursday,
March 17, 13:30. Room: Amado 719.
·
Thursday, March 17, 13:30. Room:
Amado 719.
Michael Natapov, Technion
A graph theoretic approach to graded identities
·
Thursday,
March 10, 13:30. Room: Amado 719.
Uniform cell decomposition and applications to Chevalley
groups.
This is joint work with Mark Berman, Uri Onn, and Pirita Paajanen.
We express definable integrals uniformly over nonArchimedean
local fields extending results of Pas, Macintyre, and DenefLoeser. We apply this to integrals over integral padic points of Chevalley groups,
in particular proving that zeta functions counting conjugacy
classes or dimensions of Hecke algebras in congruence
quotients depend only on the size of the residue class field, for sufficiently
large residue characteristic.
We also show that certain zeta functions associated with Chevalley groups defined over a nonArchimedean local field
are given by a rational function satisfying a functional equation, and we
obtain the same dependence only on the size of the residue class field.
· Thursday,
March 3, 13:30, Room Amado
719.
Upper bound for the size of stabilizers in finite simple or
transitive permutation groups.
1) Let G be a finite transitive permutation group on a finite set S.
Let A be a nonempty subset of S and denote the pointwise
stabilizer of A in G by C(A). We'll find an upper
bound for C(A) which depends on A
(and of course depends also on G and S)
Same bound holds if G is a simple group .
2) Let G be a finite
transitive permutation group on a finite set S. Let A be a nonempty subset of S
and denote the pointwise stabilizer of A in G by C(A). Let T be a normal subset of G not containing 1. Let U
be a Ginvariant subset of S. We'll demonstrate a connection between the
relative part of A in U and the relative part of C(A)
in T.
Same connection holds if G is a simple group .
Organized
by: David Chillag
__________________________________________________________________________________
· Thursday, Jan 20,
12:30, Room: Amado 719
JungMiao Kuo,
National Chung Hsing University, Taiwan
A universality property of the
Clifford algebras of ternary cubic forms
Abstract: In this talk, we introduce an algebra associated to a cubic curve
*C* defined over
a field *F* of characteristic not two or three. This algebra
is Azumaya of
rank 9 and its center is the affine coordinate ring of an
elliptic curve, namely the Jacobian of the cubic curve *C*. The induced
function from the group of *F*rational
points on the Jacobian into the
Brauer group of *F* is a group
homomorphism with image precisely the
relative Brauer
group of classes of central simple *F*algebras split by the
function field of *C*. Also, this algebra
is split if and only if the cubic
curve *C* has an *F*rational point.
Finally, we present a universality
property of the family of such algebras.
These results generalize Haile's
work on the Clifford algebra of a
binary cubic form.
· Thursday, Jan 13, 12:30, Room: Amado 719
RonyBitan, Technion
The discriminant of an algebraic torus.
Abstract:
Let T be an algebraic torus defined over a local field K with
discrete valuation ring O and finite residue field k. As T(K) is locally compact,
it admits a multiplicative Haar measure which is unique up to a scalar
multiplication.
Let T(O) be the maximal compact subgroup
of T(K). Its volume with respect to such a normalized measure is an arithmetic
invariant which plays an important role in the quasidiscriminant formulated by
Ono and Shyr for tori defined over global fields. We use the standard integral
model defined by Voskresenskii and its reduction over
k in order to describe the structure of T(O) and
measure it.
Due to a construction of Kottwitz, the result is expressed in terms of the cocharacter group of T.
· Thursday,
Jan 6, 12:30, Room: Amado 719
Lucas Fresse, Hebrew University
Smooth orbital varieties and orbital varieties
with a dense Borbit
Abstract: To a nilpotent element x
in a reductive Lie algebra, one can attach several algebraic varieties which
play roles in geometric representation theory: its nilpotent orbit; the
intersection of its nilpotent orbit with the nilradical
of a Borel subalgebra (the
irreducible components of this intersection are called orbital varieties); the
fiber over x of the Springer resolution. There is a close relation between the
Springer fiber over x and the orbital varieties attached to x. In this talk, we
rely on this relation in order to study two properties of orbital varieties:
the smoothness, and the property to contain a dense Borbit. We concentrate on
the type A. We provide several results which suggest that the two mentioned
properties are related. This is a joint work with Anna Melnikov.
· Thursday, Dec 30, 12:30, Room: Amado 719
Yevgeny Musicantov, Technion
On p^n dimensional semisimple
Abstract: Let G be a finite group
and let D be a fusion category. Consider the following situation, C is a
Gextension of D and M is an indecomposable Cmodule category, such that,
its restriction to D remains indecomposable. We prove that the dual C*_M is a
Gequivariantization of D*_M. We apply the above
theorem to the classification problem of the p^ndimensional
semisimple Hopf algebras.
We completely classify the p^3dimensional Hopf
algebras, recovering the result of Masuoka.
· Thursday, Dec 23, 12:30, Room: Amado 719
Dmitry Kerner, U. Toronto
On local determinantal
representations of hypersurfaces, aka maximally Cohen
Macaulay modules
Abstract:
Let
M be a square matrix whose entries are series in several variables. M is
considered up to multiplication by locally invertible matrices from both sides.
In classical algebraic geometry such an object is called the local determinantal representation of the corresponding hypersurface {det(M)=0}. In commutative algebra such objects are called
maximally Cohen Macaulay modules. In local geometry they correspond to specific
elements of the local class group.
When the hypersurface singularity is of ADE type the
classification
of such modules is finite. For higher singularities such modules have
moduli and the classification problem is wild.
I will present various points of view on these objects. Time permitting I will formulate some recent results: various necessary and
sufficient criteria of decomposability of modules over the hypersurfaces.
· Thursday, Dec 16, 12:30, Room: Amado 719
Gary Zeitz, U. Oregon
Unipotent elements in algebraic groups and nilpotent
elements in
their Lie algebras
· Thursday, Dec 9, 12:30, Room: Amado 719
Zur Itzhakian, Tel
Aviv U.
A
Glimpse at Supertropical Algebra
Abstract:
The
objective of the talk is to introduce an algebraic structure
rich enough to support algebraic formulations of properties
of
tropical geometry. In
order to bypass the lack of additive
inverses in idempotent semirings,
we consider a ``cover'' semiring
structure that has a distinguished ``ghost ideal'' taking
the
place of the zero element in many of the theorems.
This
structure permits a systematic development both of polynomial
algebra and of matrix algebra, yielding direct analogs to many
results and notions from commutative algebra. These
establish the
connection between algebra and geometry in the supertropical
setting, leading also to a natural approach to linear
algebra and
semigroup representations.
· Thursday, Dec 2, 12:30, Room: Amado 719
Darrell Haile, U. Indiana and Technion
The
Clifford algebra of a quartic curve of genus one
Abstract:
This
is joint work with Ilseop Han. For each irreducible
quartic f over a
field k, we construct a kalgebra A_f
associated to the hyperelliptic
affine curve C:y^2=f(x). We prove that A_f
has many interesting
properties. For example it is an Azumaya algebra of rank 4 over its center
and its center is the coordinate ring of the affine elliptic
curve E
related to the Jacobian of C. Each
simple image of A_f is a quaternion
algebra. The simple images with center k then come from the
rational
points on E and the resulting function from the group of
rational points
on E to the Brauer group of k is a
group homomorphism. We also prove that
A_f is split if and only if the curve C has a krational
point.
· Thursday, Nov 18, 12:30, Room: Amado 719
Eli Matzri (Technion)
Non cyclic algebras with ncentral elements.
Abstract:
Let D be a division algebra ( i.e. every non zero element
of D is invertible) of finite dimension over it's center, F.
It is known that [D:F]=n^2 for some natural number n
which is called the degree of D.
For every element d in D, F[d] is a field. We call D a cyclic algebra if it has
a subfield of dimension n which is Galois over F with group C_n.
More generally we call D a crossed product with respect to a group G is D has a
subfield Galois over F with group G.
Cyclic algebras are very important in the theory of finite dimensional division
algebras due to a wonderful theorem of Merkurjev and Suslin stating that if the center contains enough roots of
unity M_k(D) for some natural k is isomorphic to the
tensor product of cyclic algebras. For D of degree a prime p, it is a well
known theorem of Albert that says that D is cyclic iff
it contains a non cental element d whoes pth power is in the
center. This theorem is clear if F contains a pth
root of unity but is true even if it does not.
P.Roquette asked if Alberts criterion is true even for non prime degrees,
and gave a counter example in degree four.
In a recent work with L.H. Rowen and U. Vishne we give examples of algebras D, of degree p^2 and a
non central element whose p^2 power is in the center but D is not cyclic
thus showing Alberts criterion for cyclicity is false for algebras of degree p^2 for
every prime p.
· Thursday, Nov 4, 12:30, Room: Amado 719
Prof. Antonio Giambruno (
Polynomial
identities and codimension growth of algebras
Abstract:
The
determination of the precise polynomial identities of a given PI
algebra (and even a generating set) seems to be an extremely hard
problem in general and so people tried, instead, to estimate the
"number" of
identities or what is better behaved, the number of "non
identities" of a certain
degree of a given algebra. In the lecture I will explain the essence
of the codimension
theory and present results for different type of PI algebras
as associative algebras, $G$graded algebras, Lie algebras, arbitrary nonassociative.
· Thursday, Oct 28, 12:30, Room: Amado 719
Shifra Reif (Weizmann Institute of
Science)
Denominator
identities for Lie superalgebras
Abstract:
In
1972 Macdonald generalized the Weyl denominator
identity to affine root systems. The simplest example of these identities
turned out to be the famous Jacobi triple product identity. In 1994 V. G. Kac and M. Wakimoto stated an
analog for some affine Lie superalgebras and showed
that it has applications in number theory. We provide identities for the rest
of the (nontwisted) affine Lie superalgebras and
deduce the Jacobi formula for counting the number of presentations of an
integer as a sum of 8 squares (joint with M. Gorelik).
· Thursday, Oct 21, 12:30, Room: Amado 719
Crystal
Hoyt (Bar Ilan University)
Good
gradings of basic Lie superalgebras
Abstract: Zgradings of
finitedimensional simple Lie algebras were classified by V.G. Kac and A.G. Elashvili in
2005. This problem arose in connection
to Walgebras. We will discuss the classification of good Zgradings
for basic Lie superalgebras. A finitedimensional
simple Lie superalgebra g=g_0 + g_1 is called basic
if g_0 is a reductive Lie algebra and there exists a nondegenerate
invariant supersymmetric bilinear form on g. Basic Lie superalgebras
were classified by V.G. Kac in 1977. These are:
sl(mn) : m not equal
to n, psl(nn), osp(m2n), D(2,1,a), F(4) and G(3).
A
Zgrading g= \sum_{j in Z} g(j) is called good if there exists e in g_0(2) such
that the map (ad e):g(j)>g(j+2) is injective for j less than or equal to
1 and is surjective for j greater than or equal to
1. If e in g_0 belongs to an sl(2)triple {e,f,h} where [e,f]=h, [h,e]=2e and [h,f]=2f, then the
Zgrading of g given by the eigenspaces of (ad h) is
a good Zgrading for e, and is called a Dynkin
grading.
All good
gradings of the exceptional Lie superalgebras,
F(4), G(3), and D(2,1,a), are Dynkin
gradings. The good Zgradings
of sl(mn), psl(nn)
and osp(m2n) are classified using pyramids,
analogously to the Lie algebra setting.
__________________________________________________________________________________
Spring
semester 2010
· Monday, June 21, 13:30, Room: TBA
Lior BarySoroker (IEM, Universität DuisburgEssen)
Irreducible
values of polynomials
Abstract: Does there exist a polynomial
f(X) such that all polynomials
f(X), f(X)+1, f(X)+2, ..., f(X)+285 are
irreducible? Clearly the
answer depends on the field the coefficients are
taken from. We will
discuss a generalization of this problem (aka Schinzel's hypothesis H
for polynomial rings), some recent results, and the connection with
Hilbert's irreducibility theorem.
·
Thursday, June 3, 13:30, Room: Amado 919
Eyal Subag (Technion)
InonuWigner
contractions as direct limits
Abstract: It is often the case that
one physical theory with a symmetry group approximates in a certain limit
another physical theory which also possesses a symmetry group. In a famous
paper of Inonu and Wigner from 1953 a new notion of contraction of Lie algebras
and their representations
was used to construct several such examples. While
the process of contracting Lie Algebras was completely rigorous and clear, the
process
of contraction of representations was very
sketchy and involved various kinds of limit formulas.
In my
talk I will describe a new definition of contraction of Lie algebra
representations in which the representation space of the "limit"
representation is a direct limit of a certain family of linear
spaces. Using this definition I will explain the various
formulas of contraction of Lie algebra representations which were given by
Inonu and Wigner. If time permits I shall also describe some new examples.
·
Monday, May 31, 14:30, Room: Amado 815 (SPECIAL
SEMINAR—NOTE CHANGE OF TIME AND PLACE)
Ehud
Meir (Technion)
Wall's complex, projective resolutions and
complexity of modules.
Abstract:
Let G be a finite group, let k be a field of prime
characteristic, and let M be a finitely generated kG
module. The
complexity of M is a numerical invariant which measures how far M is
from being projective (for example M has complexity 0 if and only if
M is projective).
Like many other homological properties, the complexity of M can be
detected over elementary abelian subgroups of G (This
is the content
of Alperin Evens Theorem).
In this talk we will consider a generalization of the notion of
complexity for infinite groups and arbitrary rings. By considering
finite quotients of a group G, we will be able to prove a
generalization of Alperin Evens Theorem. We will
prove that under
certain assumptions, the complexity of a module M over G can be
detected over some set of finite index subgroups of G. For the proof,
we use in a fundamental way a theorem of Serre and a
complex that was
originally constructed by Wall. We will also show how we can use our
results in order to construct resolutions for the integral special
linear groups.
·
Thursday, May 27, 13:30, Room: Amado 919
Ori Parzan (Hebrew U)
Isospectrality and representations of finite groups
Abstract:
A simple phenomenon in finite groups stands behind several famous
constructions: arithmetically equivalent number fields (Gassman
1925), isospectral
manifolds (Sunada 1985), drums (Gordon et. al 1992)
and graphs (Brooks 1997), and Lie groups with the same Witten zeta function
(Larsen 2004). The phenomenon is this: $H$ and $H'$ being conjugate subgroups
of $G$ implies that ${\rm Ind}_{H}^{G}1\cong{\rm Ind}_{H'}^{G}1$,
but for some $G,H,H'$ the converse is false. We explain how to construct isospectral objects exploiting a more general situation:
${\rm Ind}_{R}^{G}R\cong{\rm Ind}_{H'}^{G}R'$
with $R$, $R'$ being any representations of $H$ and $H'$. The talk is intended
for general audience and assumes only basic notions of the representation
theory of finite groups (e.g. orthogonality of
irreducible characters).
This is joint work with Ram Band.
·
Thursday, April 29, 13:30, Room: Amado 919
Ehud
Meir (Technion)
On
the Hopf Schur group of a
field
Abstract:
Let k be a field. We ask what kcentral
simple algebras can we get as
quotients of Hopf algebras. We call the
corresponding subgroup of
Br(k) the Hopf Schur group of k.
This generalizes the question of what central simple kalgebras are
quotients of group algebras (the so called Schur
algebras) and the
Schur subgroup of Br(k).
Since group algebras are Hopf algebras, every
Schur algebra is a Hopf Schur algebra.
The Schur subgroup might be a very small subgroup of Br(k), and we ask
what other Hopf Schur algebras
can we find.
In this talk I will try to explain why actually every kcentral simple
algebra is a Hopf Schur algebra up to Brauer
equivalence. We shall do
so by giving a representation theoretic definition of Hopf
algebras,
and by considering forms of Hopf algebras.
All the relevant notions will be explained.
(Partially
based on a joint work with Eli Aljadeff, Juan Cuadra and
Shlomo Gelaki)
·
Thursday, April 22, 13:30, Room: Amado 919
Leonid MakarLimanov (Wayne State U.)
A
new approach to the twodimensional Jacobian
conjecture
Abstract:
In
my talk I'll introduce and discuss some properties of
a threedimensional polytope which
can be attached to a pair of
polynomials with the constant Jacobian.
This approach gives new
restrictions on a potential counterexample to the Jacobian Conjecture.
·
Thursday, April 8, 13:30, Room: Amado 919
Yuval Ginosar (Haifa U)
On
two notions due to von Neumann
·
Thursday, Mar 18, 13:30, Room: Amado 919
Adi Wolf (Technion)
Automorphisms and Forms of Path Algebras
Abstract:
Given a Galois extension K/F and a finite
dimensional Falgebra A, we consider the question
of classifying K/Fforms of A. When A=Mat_{n}(F) or
A=F^{n} these are known to be the central
simple algebras of degree n and the étale
algebras of dimension n, respectively.
We show how some algebras whose nilpotent radical is not trivial, namely
hereditary path
algebras with acyclicity restriction on the quiver,
are classified by certain H^{1}cohomology
sets and provide a combinatorial description of their K/Fforms in some cases.
This talk is based on a research thesis done under the supervision of Prof. Eli
Aljadeff.
·
Thursday, Mar 11, 13:30, Room: Amado 919
Shlomo Gelaki (Technion)
Semisimplicity in symmetric tensor categories
Abstract:
Let n be a positive integer, and let k be a
field of characteristic 0
or p > n. Let G be any group, let C = Rep(G) be the
category of finite
dimensional representations of G over k, and let V be an object of C.
Serre proved that if the nth exterior power of
V (respectively, the
nth symmetric power of V) is semisimple and the
dimension of V is not
equal in k to 2,3,...,n (respectively, n,...,2), then V is semisimple.
Moreover, Serre explained that the same results
hold for Lie algebras,
and conjectured that they should hold more generally in any symmetric
tensor category C over k.
Recently we proved a generalization of Serre's
conjecture. Namely, we
replaced Rep(G) by any symmetric tensor category C
over k, as suggested
by Serre, and moreover replaced the nth exterior
power and symmetric power
by any Schur functor
S_{\lambda} (here \lambda is a partition of n).
In my talk I will state our results and explain its proofs.
Organized by: Jack Sonn
_________________________________________________________________________________________
·
Thursday, Feb. 18, 11:30, Room: Amado 814
Emanuele Pacifici (Univ. of
Milan)
On
Zeroes of Characters of Finite Groups
·
Thursday, Jan 28, 10:30, Room: Amado 814
Hila Maayan (Technion)
On
Zeroes of Characters of Finite Groups
(The
talk will be given in Hebrew)
·
Thursday, Jan 21, 10:30, Room: Amado 814
Ido Efrat (BGU)
The
descending central sequence of an absolute Galois group
Abstract:
Given a field $F$, the descending $p$central sequence of its absolute Galois
group
gives a
We show that the Galois group of the extension $F^{(3)}$
of $F$ already encodes
much of the arithmetical structure of $F$, as well as its mod$p$ cohomology.
This is used to construct new examples of profinite
groups which are not realizable
as absolute Galois groups.
·
Thursday, Jan 14, 10:30, Room: Amado 814
Rony Bitan (Bar Ilan)
The
discriminant of an algebraic torus
Abstract:
Let K be
a global field.
The relation between arithmetic invariants of K such as the discriminant,
regulator, class number and the residue of the Dedekind Zetafunction is
wellknown. Let T be an algebraic
Ktorus.
Following Ono and Shyr we use analogues of these
invariants of T to
express this relation for T, by gathering the information in all local
fields. Locally, for any prime p, we compute the discriminant of T using
an integral scheme of T, namely the Neron model of T.
This local information is expressed in terms of the group of cocharacters of
T over K_p.
·
Thursday, Dec 31, 10:30, Room: Amado 814
David
Chillag (Technion)
Rationality
of finite groups and primes dividing their orders
Abstract:
A rational element of a group is an element for
which all the generators of the cyclic
subgroup it generates, are conjugate. A group is
rational if all its elements are rational.
We discuss rational groups and variations of
such, in particular the prime divisors of their orders.
· Thursday, Dec 24, 10:30, Room: Amado
814
Pavel Shumyatsky (Univ. of
Brazilia)
On
elements of primepower index in finite groups
· Thursday, Dec 10, 10:30, Room:
Amado 814
Alexei BelovKanel (Bar Ilan U.)
The Specht problem in positive characteristic
Abstract:
The Specht problem for an algebra A asks whether the ideal of
polynomial
identities of A is finitely based (to be defined in the
lecture).
In the
lecture I will discuss the Specht problem where A is
an associative
algebra over a field of positive characteristic. The main
tools used in
the solution are taken from representation theory
of associative algebras.
In
particular one needs to consider the interaction between the semisimple
part and the radical of the algebra. Similar
problems may be posed for
other structures as groups and Lie algebras.
· Monday, Nov 30, 10:30, Room: Amado
719
Enric Ventura (Universitat
Politecnica de Catalunya)
Whitehead
Minimization in Polynomial Time
Abstract:
The
Whitehead minimization problem consists in finding a minimum size element in
the automorphic orbit of a word, a cyclic word or a
finitely generated subgroup in a finite rank free group. We give the first
fully polynomial algorithm to solve this problem,
that is, an algorithm that is polynomial both in the length of the input word
and in the rank of the free group. Earlier (classical) algorithms had an
exponential
dependency in the rank of the free group. It follows that
the primitivity problem (to decide whether a word is
an element of some basis of the free group) and the free factor problem can
also be solved in polynomial time.
· Thursday, Nov 26, 10:30, Amado 814
Eli Matzri (Technion)
A birational
description of the Brauer Severi
variety of a CSA in terms of
the reduced norm.
Abstract:
Let A be
a central simple algebra over a field F. To every such A one can associate an
algebraic variety called the Brauer Severi variety of A and denoted BSV(A),
which is the variety of minimal left ideals of A. Two central simple algebras A
and B are isomorphic iff
they have isomorphic BSVs.
It was conjectured by Amitsur that A and B of the same dimension have birational
BSV iff they generate the same subgroup of Br(F).
Amitsur proved that if BSV(A)
and BSV(B) are birational then
<[A]>=<[B]>. The other direction is still open in general but is
known for some specific cases. We give a birational
description of BSV(A) in terms of the
reduced norm, and use it to reprove Amitsur's
conjecture in the case of symbol algebras.
Then we will discuss a possible birational description
of generalized
BSVs.
· Thursday, Nov 19, 10:30, Amado 814
Michael Natapov (Technion)
Graded
identities of matrix algebras with a fine group grading
Abstract:
Let A be
a ring of nbyn matrices over an algebraically closed field. Any group grading
on A is known to be induced from an elementary and a fine gradings.
While there is a wide literature on elementary gradings
and the corresponding graded polynomial identities, the fine gradings and their polynomial identities are less studied.
In particular, a minimal generating set for the ideal of Ggraded polynomial
identities of A with a fine Ggrading is known in a case of G abelian only. In a joint work with Darrell Haile we
consider a certain family of nonabelian groups G,
and study the Ggraded polynomial identities on A with a fine Ggrading in a
search for the minimal generating sets of the ideal of identities.
· Thursday, Nov 12, 10:30,
Amado 919
Michael
Schein (Bar Ilan)
On
irreducible supersingular mod p representations of
GL_2(F)
Abstract:
Let F be
a finite extension of Q_p. The mod p local Langlands correspondence should be a natural bijection between ndimensional mod p representations r of
the absolute Galois group of F and a certain class of irreducible mod p representations
L(r) of GL_n(F). Irreducible Galois representations correspond to supersingular representations of GL_n(F).
The mod
p representation theory of GL_n(F) is poorly understood, and, apart from some special
cases, few irreducible supersingular representations
have been constructed. One can use generalizations of Serre's
conjecture to specify what the socle of the
restriction of L(r) to a maximal compact subgroup should be. We will show
that supersingular representations of GL_2(F) with such socles are
generically irreducible and discuss work in progress to construct families of
such representations. The relevant notions will be defined.
· Thursday, Nov 5, 10:30, Amado 919
Daniele d'Angeli (Technion)
Selfsimilar
groups and finite Gelfand pairs
Abstract:
The
class of self similar groups has been largely studied in the last decades
providing interesting examples of groups with special and exotic properties.
These groups are groups of automorphisms of rooted qary trees. The theory of finite Gelfand
pairs appears in this context when we consider the action of a (level
transitive) selfsimilar group on the levels of the tree and the stabilizer of
a fixed vertex. In this talk I will discuss the example of the so called
Basilica group, showing that the action on each level gives rise to a symmetric
Gelfand pair.
· Thursday, Oct 29, 10:30, Amado 919
John Meakin (University of Nebraska)
Submonoids of Groups
Abstract:
Semigroups
that embed in groups have received considerable attention in the classical and
modern literature in both semigroup theory and group
theory. For example, one obtains useful information about large classes of
groups such as Adian groups, braid groups and Artin groups by studying their positive submonoids.
I will discuss some of these results as well as some recent work and unsolved
problems linking algorithmic problems for inverse monoids
to corresponding problems for submonoids of groups.
· Thursday, Oct 22, 10:30,
Amado 919
Danny Neftin (Technion)
On semiabelian groups and the minimal ramification problem
Abstract:
Let $p$
be a prime number and $G$ a $p$group of rank $r$, i.e. $G$ is generated by $r$
elements and not less. It is conjectured that $G$ can be realized over
$Q$ with exactly $r$ ramified primes. Kisilevsky and Sonn showed the conjecture holds for a certain family of semiabelian groups and asked whether this family is the
family of all semiabelian groups. This question was
answered positively. We shall discuss the proof and how it can be used to
simply the proof of Kisilevsky and Sonn and extend it. (Joint work with Hershy
Kisilevsky and Jack Sonn)
Organized by: Jack Sonn