Page 277 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 277
CONFIGURATIONS (MS-81)
Jordan schemes
Sven Reichard, sven.reichard@tu-dresden.de
Dresden International University, Germany
Association schemes (also called homogeneous coherent configurations) first appeared in Statis-
tics in relation to the design of experiments, due to the efforts of Bose and his collaborators.
These systems of binary relations defined on the same set, color graphs in other terms, give
rise to certain matrix algebras, not necessarily commutative. In this way a bridge is formed
between combinatorics and algebra, in particular with permutation groups.
In statistial applications we want the relations to be symmetric, while the product of sym-
metric matrices is not symmetric in general. Therefore Bailey and Cameron, following Shah
(from the same school as Bose) suggested to replace the matrix product AB by the Jordan prod-
uct A∗B = (AB +BA)/2, which is commutative but not necessarily associative. The resulting
structures are called Jordan schemes.
Given any association scheme, its symmetrization is a Jordan scheme. This led Peter
Cameron to the following question: Do all Jordan schemes arise in this way, or do there ex-
ist “proper” Jordan schemes?
We gave a positive answer to this question by constructing a first proper Jordan scheme on 15
points using so-called Siamese color graphs, investigated earlier by our group. First elements of
the theory of these structures were established; a few infinite classes of proper Jordan schemes
were discovered. Moreover an efficient computational criterion for recognizing proper Jordan
schemes is given, based on the classical Weisfeiler-Leman stabilization.
The current talk mainly focuses on the computer search for proper Jordan schemes, which
is based on algorithmic ideas of Hanaki and Miyamoto, who enumerated all small association
schemes. It turns out that the initial example is indeed the smallest. Besides it, up to isomor-
phism three more examples on less than 20 points were discovered. Each such small example
of orders 15, 16 and 18 can be constructed from a suitable association scheme by a certain
switching operation.
For more on the background of Jordan schemes see Cameron’s blog:
https://cameroncounts.wordpress.com/2019/06/28/proper-jordan-schemes-exist/
This is part of a joint project with Misha Klin and Misha Muzychuk from Ben-Gurion
University, Beer Sheva.
Taxonomy of Three-Qubit Doilies
Metod Saniga, msaniga@astro.sk
Slovak Academy of Sciences, Slovakia
Coauthors: Henri de Boutray, Frédéric Holweck, Alain Giorgetti
We study doilies (i. e., W (3, 2)’s) living in W (5, 2), when the points of the latter space are
parametrized by canonical three-fold products of Pauli matrices and the associated identity
matrix (i. e., by three-qubit observables). Key characteristics of such a doily are: the number
of its negative lines, distribution of types of observables, character of the geometric hyperplane
the doily shares with the distinguished (non-singular) quadric of W (5, 2) and the structure of its
Veldkamp space. W (5, 2) is endowed with 90 negative lines of two types and its 1344 doilies
fall into 13 types. 279 out of 480 doilies with three negative lines are composite, i. e. they all
275
Jordan schemes
Sven Reichard, sven.reichard@tu-dresden.de
Dresden International University, Germany
Association schemes (also called homogeneous coherent configurations) first appeared in Statis-
tics in relation to the design of experiments, due to the efforts of Bose and his collaborators.
These systems of binary relations defined on the same set, color graphs in other terms, give
rise to certain matrix algebras, not necessarily commutative. In this way a bridge is formed
between combinatorics and algebra, in particular with permutation groups.
In statistial applications we want the relations to be symmetric, while the product of sym-
metric matrices is not symmetric in general. Therefore Bailey and Cameron, following Shah
(from the same school as Bose) suggested to replace the matrix product AB by the Jordan prod-
uct A∗B = (AB +BA)/2, which is commutative but not necessarily associative. The resulting
structures are called Jordan schemes.
Given any association scheme, its symmetrization is a Jordan scheme. This led Peter
Cameron to the following question: Do all Jordan schemes arise in this way, or do there ex-
ist “proper” Jordan schemes?
We gave a positive answer to this question by constructing a first proper Jordan scheme on 15
points using so-called Siamese color graphs, investigated earlier by our group. First elements of
the theory of these structures were established; a few infinite classes of proper Jordan schemes
were discovered. Moreover an efficient computational criterion for recognizing proper Jordan
schemes is given, based on the classical Weisfeiler-Leman stabilization.
The current talk mainly focuses on the computer search for proper Jordan schemes, which
is based on algorithmic ideas of Hanaki and Miyamoto, who enumerated all small association
schemes. It turns out that the initial example is indeed the smallest. Besides it, up to isomor-
phism three more examples on less than 20 points were discovered. Each such small example
of orders 15, 16 and 18 can be constructed from a suitable association scheme by a certain
switching operation.
For more on the background of Jordan schemes see Cameron’s blog:
https://cameroncounts.wordpress.com/2019/06/28/proper-jordan-schemes-exist/
This is part of a joint project with Misha Klin and Misha Muzychuk from Ben-Gurion
University, Beer Sheva.
Taxonomy of Three-Qubit Doilies
Metod Saniga, msaniga@astro.sk
Slovak Academy of Sciences, Slovakia
Coauthors: Henri de Boutray, Frédéric Holweck, Alain Giorgetti
We study doilies (i. e., W (3, 2)’s) living in W (5, 2), when the points of the latter space are
parametrized by canonical three-fold products of Pauli matrices and the associated identity
matrix (i. e., by three-qubit observables). Key characteristics of such a doily are: the number
of its negative lines, distribution of types of observables, character of the geometric hyperplane
the doily shares with the distinguished (non-singular) quadric of W (5, 2) and the structure of its
Veldkamp space. W (5, 2) is endowed with 90 negative lines of two types and its 1344 doilies
fall into 13 types. 279 out of 480 doilies with three negative lines are composite, i. e. they all
275
