Some properties of an Euclidean Jordan algebra spanned by a symmetric association scheme

2020 ◽  
Author(s):  
Lúıs António de Almeida Vieira
Keyword(s):  
Jordan Algebra ◽  
Association Scheme ◽  
Euclidean Jordan Algebra ◽  
Symmetric Association Scheme

scholarly journals A Spectral Equivalent Condition of the $P$-Polynomial Property for Association Schemes

10.37236/4423 ◽  
2014 ◽  
Vol 21 (3) ◽  
Author(s):  
Hiroshi Nozaki ◽  
Hirotake Kurihara
Keyword(s):  
Association Scheme ◽  
Association Schemes ◽  
Equivalent Condition ◽  
Polynomial Property ◽  
Equivalent Conditions ◽  
Symmetric Association Scheme

We give two equivalent conditions of the $P$-polynomial property of a symmetric association scheme. The first equivalent condition shows that the $P$-polynomial property is determined only by the first and second eigenmatrices of the symmetric association scheme. The second equivalent condition is another expression of the first using predistance polynomials.

scholarly journals An analog of Thompson's triangle inequality in Euclidean Jordan algebras

2021 ◽  
Vol 37 (37) ◽  
pp. 156-159
Author(s):  
Jiyuan Tao
Keyword(s):  
Linear Algebra ◽  
Jordan Algebra ◽  
Triangle Inequality ◽  
Short Note ◽  
Direct Proof ◽  
Case Analysis ◽  
Euclidean Jordan Algebra ◽  
Jordan Algebras ◽  
Euclidean Jordan Algebras

In a recent paper [Linear Algebra Appl., 461:92--122, 2014], Tao et al. proved an analog of Thompson's triangle inequality for a simple Euclidean Jordan algebra by using a case-by-case analysis. In this short note, we provide a direct proof that is valid on any Euclidean Jordan algebras.

A REPRESENTATION THEOREM FOR LYAPUNOV-LIKE TRANSFORMATIONS ON EUCLIDEAN JORDAN ALGEBRAS

2013 ◽  
Vol 15 (04) ◽  
pp. 1340034 ◽  
Author(s):  
JIYUAN TAO ◽  
M. SEETHARAMA GOWDA
Keyword(s):  
Jordan Algebra ◽  
Linear Transformation ◽  
Representation Theorem ◽  
Elementary Proof ◽  
Euclidean Jordan Algebra ◽  
Symmetric Cone ◽  
Jordan Algebras ◽  
Euclidean Jordan Algebras

A Lyapunov-like (linear) transformation L on a Euclidean Jordan algebra V is defined by the condition [Formula: see text]where K is the symmetric cone of V. In this paper, we give an elementary proof (avoiding Lie algebraic ideas and results) of the fact that Lyapunov-like transformations on V are of the form La + D, where a ∈ V, D is a derivation, and La(x) = a ◦ x for all x ∈ V.

scholarly journals Symmetric geodesics on conformal compactifications of Euclidean Jordan algebras

1999 ◽  
Vol 59 (2) ◽  
pp. 187-201
Author(s):  
Sang Youl Lee ◽  
Yongdo Lim ◽  
Chan-Young Park
Keyword(s):  
Jordan Algebra ◽  
Euclidean Jordan Algebra ◽  
Jordan Algebras ◽  
Closed Geodesics ◽  
Shilov Boundary ◽  
Torus Knots ◽  
Euclidean Jordan Algebras ◽  
Real Matrices

In this article we define symmetric geodesies on conformal compactifications of Euclidean Jordan algebras and classify symmetric geodesics for the Euclidean Jordan algebra of all n × n symmetric real matrices. Furthermore, we show that the closed geodesics for the Euclidean Jordan algebra of all 2 × 2 symmetric real matrices are realised as the torus knots in the Shilov boundary of a Lie ball.

scholarly journals The Multiplicities of a Dual-thin $Q$-polynomial Association Scheme

10.37236/1589 ◽  
2001 ◽  
Vol 8 (1) ◽  
Author(s):  
Bruce E. Sagan ◽  
John S. Caughman, IV
Keyword(s):  
Association Scheme ◽  
Primitive Idempotents ◽  
Polynomial Association Scheme ◽  
Symmetric Association Scheme

Let $Y=(X, \{ R_i \}_{1\le i\le D})$ denote a symmetric association scheme, and assume that $Y$ is $Q$-polynomial with respect to an ordering $E_0,...,E_D$ of the primitive idempotents. Bannai and Ito conjectured that the associated sequence of multiplicities $m_i$ $(0 \leq i \leq D)$ of $Y$ is unimodal. Talking to Terwilliger, Stanton made the related conjecture that $m_i \leq m_{i+1}$ and $m_i \leq m_{D-i}$ for $i < D/2$. We prove that if $Y$ is dual-thin in the sense of Terwilliger, then the Stanton conjecture is true.

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