March  4, 2024
Erhard Neher, Kirill Zainoulline
Department of Mathematics and Statistics, University of Ottawa

Invited Abstracts

Sanghoon Baek, University of Ottawa
Annihilators of torsion of Chow groups of twisted spin flags

We discuss the relationship between basic polynomial invariants and classes of fundamental representations. This provides information on the torsion of the Grothendieck gamma ?ltration and the Chow groups of twisted spin flags.

This is joint work with Erhard Neher, Kirill Zainoulline, and Changlong Zhong.

Vladimir Chernousov
, University of Alberta
Groups of type $F_4$ over regular local rings

In the talk we discuss the Grothendieck-Serre conjecture and Purity conjecture for torsors of type $F_4$.

Alberto Elduque,
Universidad de Zaragoza
Gradings on the octonions and the Albert algebra

Given a grading $\Gamma: A=\oplus_{g\in G}A_g$ on a nonassociative algebra $A$ by an abelian group $G$, we have two subgroups of the group of automorphisms of $A$: the automorphisms that stabilize each homogeneous component $A_g$ (as a subspace) and the automorphisms that permute the components. By the Weyl group of $\Gamma$ we mean the quotient of the latter subgroup by the former. In the case of a Cartan decomposition of a semisimple complex Lie algebra, this is the automorphism group of the root system, i.e., the so-called extended Weyl group. A grading is called fine if it cannot be refined.

The fine gradings on the octonions and the Albert algebra over an algebraically closed field (of characteristic different from 2 in the case of the Albert algebra) will be described, as well as the corresponding Weyl groups.

John Faulkner, University of Virginia
Weyl images of Kantor Pairs

$BC_1$-graded Lie algebras correspond to Kantor pairs (including Jordan pairs) via the Kantor construction. Similarly, $BC_2$-graded Lie algebras correspond to Kantor pairs with so called short Peirce (SP) gradings. The Weyl group of $BC_2$ acts on the $BC_2$-gradings and hence on the Kantor pairs with SP gradings. Sometimes, a Weyl image of a Jordan pair is not Jordan. For finite dimensional simple Lie algebras, the Kantor pairs with SP gradings are determined by certain pairs of subsets of the Dynkin diagram, and it is easy to compute how these subsets change for a Weyl image. Also, the fundamental reflection of a Kantor pair with

K-dimension 1 can be constructed by a doubling process on certain Jordan pairs.

This is joint work with Bruce Allison and Oleg Smirnov.

Skip Garibaldi, Emory University
Did a 1-dimensional magnet detect $E_8$?

You may have heard the rumors that $E_8$ has been detected in a laboratory experiment involving a 2-centimeter-long magnet. This is fascinating, both because of $E_8$'s celebrity and because it is hard to imagine a realistic experiment that could directly observe a 248-dimensional object such as $E_8$. The purpose of this talk is to address some of the natural questions, such as: What exactly did the physicists do? How was $E_8$ involved? Does it make sense to say that they detected $E_8$? Was it the $E_8$ root system, a Lie algebra, or a Lie group? And why $E_8$ and not some other type, like $E_7$ or $E_6$? This talk is based on a joint paper with David Borthwick.

Nikita Karpenko,
Universite Pierre et Marie Curie / Institut de Mathematiques de Jussieu
On $p$-generic splitting varieties

The talk is based on a joint work with A. S. Merkurjev.

Given a prime integer $p$, we detect a class of $p$-generic splitting varieties $X$ of a symbol in the Galois
cohomology of a field $F$ such that for any equidimensional variety $Y$, the change of field homomorphism $\mathrm{CH}(Y)\to\mathrm{CH}(Y_{F(X)})$ of Chow groups with coefficients in integers localized at $p$
is surjective in codimensions $< (\dim X)/(p-1)$. This applies to projective homogeneous varieties of type $F_4$ and $E_8$.

Max Knus, ETH Zurich
Triality over arbitrary fields and over $\mathbb F_1$

This is a report on work with V. Chernousov and J.-P. Tignol. Clasical triality occurs in two different settings, as a geometric property of $6$-dimensional quadrics and as outer automorphisms of order $3$ of simple groups of type $D_4$. As already observed by \'E. Cartan, triality is closely related to octonions. In the first part of the talk we show that, over arbitrary fields, the classification of trialities is equivalent to the classification of certain $8$-dimensional composition algebras. In the second part we discuss in a parallel way triality over Tits' field with one element.

Mark MacDonald,
University of British Columbia
Essential dimension of the exceptional groups

In this talk I will survey what is known about the essential dimension and essential $p$-dimension for each of the exceptional groups, including some new upper bounds for $F_4$, and simply connected $E_6$ and $E_7$. I will explain some of the techniques used to find these bounds, including an analysis of their small linear representations.

Tom De Medts,
Ghent University (Belgium)
Exceptional Moufang quadrangles and $J$-ternary algebras

Moufang polygons have been introduced by Jacques Tits in order to describe the linear algebraic groups of relative rank two,
and in fact, one of the main motivations is precisely to get a better understanding of the exceptional groups.
In particular, there are certain rank two forms of groups of type $E_6$, $E_7$ and $E_8$, for which the corresponding Moufang quadrangles have been described by Richard Weiss in terms of so-called quadrangular algebras.
The explicit construction, however, is rather wild, and requires a very careful coordinatization of certain vector spaces of dimension
$8$, $16$ and $32$, respectively.

We have found a conceptual way of constructing these quadrangular algebras, starting from $J$-ternary algebras,
where $J$ is a Jordan algebra of capacity two. In the case of the Moufang quadrangles of type $E_8$, for instance, this construction involves the tensor product of two octonion division algebras.

We believe that, in fact, every Moufang quadrangle defined over a field of characteristic different from $2$ can
be obtained in a similar fashion.

This is joint work with Lien Boelaert.

Raman Parimala,
Emory University
R-triviality of certain simply connected groups of type $E_8$

We discuss the $R$-triviality of certain simply connected groups of type $E_8,2^{6,6}$. This leads to the affirmative solution to Kneser-Tits problem for this case.

This is a joint work with J.-P. Tignol and R. Weiss.

Holger P. Petersson, Fakult\"at f\"ur Mathematik und Informatik \\FernUniversit\"at in Hagen
Albert algebras

Albert algebras belong to a wider class of algebraic structures called \emph{Jordan algebras}. Originally designed in the early
nineteen-thirties as a tool to understand the foundations of quantum mechanics, Jordan algebras in the intervening decades have grown
into a full-fledged mathematical theory, with profound applications to various branches of algebra, analysis, and geometry. Albert
algebras, along with their natural allies called \emph{cubic Jordan algebras}, form an important subclass whose significance comes to
the fore through the connection with exceptional algebraic groups and Lie algebras. In order to exploit this connection to the
fullest, a thorough understanding of cubic Jordan algebras in general, and of Albert algebras in particular, is indispensable. The
primary purpose of my lecture will be to lay the foundations for such an understanding. More specifically, it will be shown that the
main concepts of the theory can be investigated over arbitrary commutative rings. Moreover, a novel approach to the two Tits
constructions of cubic Jordan algebras will be presented that works in this generality and yields new insights even when the base ring
is a field. We then proceed to describe the basic properties of Albert \emph{division} algebras, with special emphasis on their
(cohomological) invariants. The lectures conclude with stating and discussing a number of open problems.

Arturo Pianzola, Alberta/Mathematical Sciences
Serre's Conjecture II, Dessins d'Enfants and Lie algebras of type $D_4$

We will describe how Serre's Conjecture II and Grothendieck's Dessins d'Enfants can be used to classify Lie algebras of type $D_4$ over complex Laurent polynomial rings in two variables and the complex projective line minus three points respectively.

Anne Quéguiner-Mathieu, Université Paris 13
Applications of triality to orthogonal involutions in degree $8$

Among Dynkin diagrams, the diagram of type $D_4$ is specific in that it does admit automorphisms of order $3$.
The corresponding simply connected algebraic group is the cover ${\mathrm {Spin}}_8$ of the special orthogonal group ${\mathrm {SO}}_8$. Inner twisted forms of this group can be viewed as the the ${\mathrm {Spin}}$ group of some algebraic structure, namely an $8$-dimensional quadratic form, of even, more generally, a degree $8$ algebra with orthogonal involution. Because of triality, those degree $8$ algebras with involution actually come by triple. Hence triality sheds a particular light on the study of involutions in degree $8$.
The talk will describe several concrete applications of this fact. For instance, we provide explicit examples of non isomorphic involutions that become isomorphic after scalar extension to a generic splitting field of the underlying algebra.

Richard M. Weiss, Tufts University
Moufang Polygons

A generalized polygon is a bipartite graph whose girth equals twice its diameter. A generalized polygon is the same thing as a spherical
building of rank~2. Generalized polygons are too numerous to classify, but Tits observed that the generalized polygons that occur as residues
of thick irreducible spherical buildings of rank at least~3 as well as the generalized polygons that are the spherical buildings associated to
absolutely simple algebraic groups of relative rank~2 all satisfy a symmetry property he called the {\it Moufang condition}. A {\it Moufang polygon} is a generalized polygon satisfying the Moufang condition. Moufang polygons were subsequently classified by Tits and myself.

The Moufang condition is expressed in terms of certain distinguished subgroups of the automorphism group called {\it root groups}. A Moufang polygon is uniquely determined by a small set of these root groups and the commutator relations between them. The classification of Moufang says that the root groups and these commutator relations are, in turn, uniquely determined by certain algebraic data. Moufang triangles (i.e. Moufang polygons of diameter~3), for example, are classified by alternative division rings and Moufang hexagons by quadratic Jordan division algebras of degree~3. The exceptional Moufang polygons---those that come from rank~2 forms of the exceptional groups---and the algebraic structures classifying them are of particular interest; these include the Moufang triangles determined by an octonion division algebra, all the Moufang hexagons and several families of Moufang quadrangles.

In my first lecture I plan to introduce Moufang polygons and give some idea of the main steps in their classification. In my second lecture, I will
focus on on the algebraic structures that arise in the context of the exceptional Moufang polygons. In the remaining lectures I will introduce buildings of arbitrary rank and attempt to indicate the central role that Moufang polygons play in Tits' classification results for spherical and affine buildings.

Contributed Talks

Hernando Bermudez, Emory University
A Unified Solution to Some Linear Preserver

We obtain a general theorem that allows the determination of the group of linear transformations on a vector space V that preserve a polyno-
mial function p on V for several interesting pairs (V; p). The proof is based on methods from the theory of semisimple linear algebraic groups, in particular a theorem of Demazure on the automorphism group of some projective varieties. Along the way we make evident the connection between the transformations that preserve the polynomial and those that preserve a set of \minimal" elements of V , a connection that had previously been observed for numerous special cases.

This is a joint work with Skip Garibaldi and Victor Larsen

Caroline Junkins, University of Ottawa
The J-invariant and Tits algebras for groups of inner type E6

A connection between the indices of the Tits algebras of a split linear algebraic group G and the degree one parameters of its motivic J-invariant was introduced by Quéguiner-Mathieu, Semenov and Zainoulline through use of the second Chern class map in the Riemann-Roch theorem without denominators. We extend their result to higher Chern class maps and provide applications to groups of inner type E6.

John Hutchens, North Carolina State University
k-involutions of Exceptional Linear Algebraic Groups
pdf available here

Timothy Pollio, University of Virginia
The multinorm principle
pdf available here

A finite extension $L/K$ of global fields is said to satisfy the Hasse norm principle if $K^{\times} \cap N_{L/K}(J_L) = N_{L/K}(L^{\times})$, where $N_{L/K} \colon J_L \to J_K$ denotes the natural extension of the norm map associated with $L/K$ to the corresponding groups of ideles. The obstruction for the Hasse norm principle, which is often nontrivial, was computed by Tate\cite{Cass} in the Galois case and by Drakokhrust\cite{Drak} in the general case. Similarly, a pair of finite extensions $L_1 , L_2$ of $K$ is said to satisfy the multinorm principle if
$$K^{\times} \cap N_{L_1/K}(J_{L_1})N_{L_2/K}(J_{L_2}) = N_{L_1/K}(L_1^{\times})N_{L_2/K}(L_2^{\times}).$$
Some sufficient conditions for the multinorm principle were given by H\"urlimann\cite{Hurlimann}, Colliot-Th\'el\`ene--Sansuc\cite{CTS}, Platonov--Rapinchuk\cite{PlR}, and Prasad--Rapinchuk\cite{PR}. These results assert the validity of the multinorm principle if the extensions are disjoint (or their Galois closures are disjoint) and one of the extensions satisfies the usual Hasse norm principle. In my joint work with Rapinchuk\cite{PoR}, we show that the multinorm principle always holds for a pair of linearly disjoint Galois extensions (even if both extensions fail to satisfy the Hasse norm principle). I will outline the proof of this theorem and gives some additional results and examples. In particular, I will discuss the situation for extensions that are not Galois or disjoint, and talk about the generalization of the multinorm principle for three or more extensions.


[1] J.W.S. Cassels, A. Frolich (Eds.), Algebraic Number Theory, Thompson
Book Company Inc., Washington D.C., 1967.
[2] Colliot-Thelene, Sansuc, Private Communication.
[3] Yu. A. Drakokhrust, On the complete obstruction to the Hasse principle,
Amer. Math. Soc. Transl.(2) 143 (1989), 29-34.
[4] Hurlimann, On algebraic tori of norm type, Comment. math. Helv. 59(1984),
[5] V.P. Platonov, A.S. Rapinchuk, Algebraic Groups and Number Theory, Academic
Press, 1994.
[6] T. Pollio, A.S. Rapinchuk, The Multinorm Principle for linearly disjoint
Galois extensions, arXiv:1203.0359v1.
[7] G. Prasad, A.S. Rapinchuk, Local-global principles for embedding of elds
with involution into simple algebras with involution, Comment. math. Helv.
85(2010), 583-645.

Igor Rapinchuk
On the conjecture of Borel and Tits for abstract homomorphisms of algebraic groups

The conjecture of Borel-Tits (1973) states that if $G$ and $G'$ are algebraic groups defined over infinite fields $k$ and $k'$, respectively, with $G$ semisimple and simply connected, then given any abstract representation $\rho \colon G(k) \to G'(k')$ with Zariski-dense image, there exists a commutative finite-dimensional $k'$-algebra $B$ and a ring homomorphism $f \colon k \to B$ such that $\rho$ can essentially be written as a composition $\sigma \circ F$, where $F \colon G(k) \to G(B)$ is the homomorphism induced by $f$ and $\sigma \colon G(B) \to G'(k')$ is a morphism of algebraic groups. We prove this conjecture in the case that $G$ is either a universal Chevalley group of rank $\geq 2$ or the group $\mathbf{SL}_{n, D}$, where $D$ is a finite-dimensional central division algebra over a field of characteristic 0 and $n \geq 3$, and $k'$ is an algebraically closed field of characteristic 0. In fact, we show, more generally that if $R$ is a commutative ring and $G$ is a universal Chevalley-Demazure group scheme of rank $ \geq 2$, then abstract representations over algebraically closed field of characteristic 0 of the elementary subgroup $E(R) \subset G(R)$ have the expected description. We also give applications to deformations of representations of $E(R).$

Changlong Zhong (University of Ottawa)
Invariant and characteristic map

In this talk we consider the characteristic map $c:S^*(\Lambda) \to CH(G/B)$ with $G$ of type B or D. We show that there is a number b_d for each d, independent on the rank of the group $G$, such that $b_d\cdot \subset I^W_a,$ where $I^W_a$ is the ideal of $S^*(\Lambda)$ generated by non-constant $W$-invariant elements. The proof uses computations of symmetric polynomials and the so-called "Ideal of generalized invariants".

This is joint work with S. Baek and K. Zainoulline.