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Questions tagged [association-schemes]

Association schemes belong to both algebra and combinatorics. They provide a unified approach to many topics, for example combinatorial designs and block codes. Association schemes also generalize groups and character theory of linear representations of groups.

Association schemes belong to both algebra and combinatorics. Indeed, in algebraic combinatorics, association schemes provide a unified approach to many topics, for example combinatorial designs and coding theory. In algebra, association schemes generalize groups, and the theory of association schemes generalizes the character theory of linear representations of groups.

See https://en.wikipedia.org/wiki/Association_scheme

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Derivation of Kravchuk polynomial identity

I am working my way through N.J.A. Sloane "An Introduction to Association Schemes and Coding Theory" and have got stuck proving the last of his identities for the Kravchuck (Krawtchouk) polynomials. The Kravchuk polynomial is defined as $$ K_k(i;n)…
brett stevens
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Proof that the conjugacy class association scheme is an association scheme

I was looking at the conjugacy class association scheme (where, given some group $G$, each conjugacy class $C_i$ gets a relation $R_i$, where $R_i=\{(x,y)|xy^{-1}\in C_i\}$), and trying to show that it's an association scheme. Defining a set…
Sam Jaques
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Does representation theory exists without Groups?

I need to know: is representation theory all about Groups? Is it necessary to be a finite group? Does representation theory exists without Groups? For example is there sample where representation is about semi-group?
IremadzeArchil19910311
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Kravchuk polynomials identity

I am trying to solve a combinatorics problem making use of Kravchuk polynomials and I am stuck with a specific sum I cannot compute. It goes like this : \begin{equation} \sum_{m=0}^{n-1}(m+1)\begin{pmatrix}n-1\\m…
Etienne
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Can we characterize the “associate classes” of a unipotent quasi-commutative quasigroup as some combinatorial design?

$I_n$ is the $n \times n$ or order $n$ identity matrix, $J_n$ is the order $n$ matrix of all ones, and $n \in \mathbb{Z}^+$. We define a Latin square $\mathcal{L_n}$ to be a set of $n$ permutation matrices $P_i$ each of order $n$ with the following…
Naiim
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In a primitive symmetric association scheme, why does $E_j$ occur in some power of $E_i$ for each $i,j$?

I am having some trouble in the proof of the Absolute Bound Condition for primitive symmetric association Schemes in the book Algebraic Combinatorics I by Bannai and Ito (Chapter 2, Section 4, Theorem 9): Let $\chi=(X,\lbrace{R_i\rbrace}_{0\leq i…
F.Tomas
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Example of non-commutative association scheme

I need an example of non-commutative association scheme of ordered 6. I tried to use the example in the book Handbook of Combinatorial Designs, Second Edition by Charles J. Colbourn‏،Jeffrey H. Dini but I could not reach the answer. I took…
user275240
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Cyclotomic character value property for Distance-regular graphs

I have read this paper. So, I am just wondering if any Distance-regular graph(DRG) has cyclotomic character value property (which means the eigenvalues of a commutative association scheme have to be cyclotomic algebraic integers) over rational…
user1992
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Composition of homomorphisms of association schemes

In Zieschang's "Theory of Association Schemes", in section 5.2, he remarks that the composition of homomorphisms is not always a homomorphism. I've been struggling to find an example of that claim. I've found certain papers which reference that…
tses
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Understanding translation association schemes

I am having trouble understanding TAS. Let $(X,+)$ be a finite abelian group. A translation assiociation scheme is an association scheme $(X, \mathbf{R})$so that for all $(x,y) \in R_i \implies (x+z,y+z) \in R_i \quad \forall i \quad \forall z\in…
Youniz
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Cyclotomic scheme is a Association scheme

I try to show that the following defines an association scheme: Let $\mathbb{F}_q$ be a field, $\omega$ a primitive element of $\mathbb{F}_q^\times$ and $s$ divides $q-1$. Define $r=\frac{q-1}{s}$, $C_0=\{0\}$, $C_1=\langle…
user239922
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$\left(n+1\right)\times \left(n+1\right)$ algebra isomorphic to Bose-Mesner algebra?

The Wikipedia article on association schemes claims regarding Bose-Mesner algebras: There is another algebra of $\left(n+1\right)\times \left(n+1\right)$ matrices which is isomorphic to ${\mathcal {A}}$, and is often easier to work with. What is…
John P.
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Clique-coclique bound in association scheme

Let $A_0=I,A_1,\dots,A_k$ be the assocation matrices of a $k-$ class association scheme $R_0,\dots,R_k$ on a set $X$. Let $K \subset \{0,1,\dots,k\}$. We say a subset $Y \subset X$ is a $K-$coclique if for any $(u,v) \in Y \times Y$, $(u,v) \not \in…
Mutasim Mim
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Characterization of Strongly Regular Graphs

I am looking for a reference in which I can find a proof of the following result. A strongly regular graph is disconnected if and only if it is a disjoint union of complete graphs $K_n$ of the same size. A strongly regular graph connected if and…
mathma
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