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| {{about|the cellular algebras of Graham and Lehrer|the cellular algebras of Weisfeiler and Lehman|association scheme}}
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| In [[abstract algebra]], a '''cellular algebra''' is a [[finite-dimensional]] [[associative algebra]] ''A'' with a distinguished '''cellular basis''' which is particularly well-adapted to studying the [[representation theory]] of ''A''.
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| == History ==
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| The cellular algebras discussed in this article were introduced in a 1996 paper of Graham and Lehrer.<ref name="grahamlehrer">{{citation|title= Cellular algebras|first1 = J.J|last1= Graham|first2= G.I.|last2 = Lehrer|journal= Inventiones Mathematicae|volume= 123|year=1996|pages=1–34}}</ref> However, the terminology had previously been used by [[Boris Weisfeiler|Weisfeiler]] and Lehman in the Soviet Union in the 1960s, to describe what are also known as [[association scheme]]s.
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| <ref>{{cite journal| last1 = Weisfeiler| first1 = B. Yu.| authorlink1 = Boris Weisfeiler| last2 = A. A. | first2 = Lehman| year = 1968| title = Reduction of a graph to a canonical form and an algebra which appears in this process| journal = Scientific-Technological Investigations| volume = 9| series = 2| pages = 12–16| language = Russian}}</ref><ref>{{cite book|first=Peter J.|last=Cameron|authorlink=Peter Cameron (mathematician)|title = Permutation Groups|publisher = [[Cambridge University Press]] | series = London Mathematical Society Student Texts (45) | year = 1999 | isbn = 0-521-65378-9}}</ref>
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| == Definitions ==
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| Let <math>R</math> be a fixed [[commutative ring]] with unit. In most applications this is a field, but this is not needed for the definitions. Let also <math>A</math> be a <math>R</math>-algebra.
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| === The concrete definition ===
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| A ''cell datum'' for <math>A</math> is a tuple <math>(\Lambda,i,M,C)</math> consisting of
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| * A finite partially ordered set <math>\Lambda</math>.
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| * A <math>R</math>-linear [[anti-automorphism]] <math>i:A\to A</math> with <math>i^2=id_A</math>.
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| * For every <math>\lambda\in\Lambda</math> a non-empty, finite set <math>M(\lambda)</math> of indices.
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| * An injective map
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| :<math>C: \dot{\bigcup}_{\lambda\in\Lambda} M(\lambda)\times M(\lambda) \to A</math>
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| :The images under this map are notated with an upper index <math>\lambda\in\Lambda</math> and two lower indices <math>\mathfrak{s},\mathfrak{t}\in M(\lambda)</math> so that the typical element of the image is written as <math>C_\mathfrak{st}^\lambda</math>.
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| and satisfying the following conditions:
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| # The image of <math>C</math> is a <math>R</math>-basis of <math>A</math>.
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| # <math>i(C_\mathfrak{st}^\lambda)=C_\mathfrak{ts}^\lambda</math> for all elements of the basis.
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| # For every <math>\lambda\in\Lambda</math>, <math>\mathfrak{s},\mathfrak{t}\in M(\lambda)</math> and every <math>a\in A</math> the equation
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| ::<math>aC_\mathfrak{st}^\lambda \equiv \sum_{\mathfrak{u}\in M(\lambda)} r_a(\mathfrak{u},\mathfrak{s}) C_\mathfrak{ut}^\lambda \mod A(<\lambda)</math>
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| ::with coefficients <math>r_a(\mathfrak{u},\mathfrak{s})\in R</math> depending only on <math>a</math>,<math>\mathfrak{u}</math> and <math>\mathfrak{s}</math> but not on <math>\mathfrak{t}</math>. Here <math>A(<\lambda)</math> denotes the <math>R</math>-span of all basis elements with upper index strictly smaller than <math>\lambda</math>.
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| This definition was originally given by Graham and Lehrer who invented cellular algebras.<ref name="grahamlehrer" />
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| === The more abstract definition ===
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| Let <math>i:A\to A</math> be an anti automorphism of <math>R</math>-algebras with <math>i^2=id</math> (just called "involution" from now on).
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| A ''cell ideal'' of <math>A</math> w.r.t. <math>i</math> is a two-sided ideal <math>J\subseteq A</math> such that the following conditions hold:
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| # <math>i(J)=J</math>.
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| # There is a left ideal <math>\Delta\subseteq J</math> that is free as a <math>R</math>-module and an isomorphism
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| ::<math>\alpha: \Delta\otimes_R i(\Delta) \to J</math>
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| ::of <math>A</math>-<math>A</math>-bimodules such that <math>\alpha</math> and <math>i</math> are compatible in the sense that
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| ::<math>\forall x,y\in\Delta: i(\alpha(x\otimes i(y))) = \alpha(y\otimes i(x))</math>
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| A ''cell chain'' for <math>A</math> w.r.t. <math>i</math> is defined as a [[direct sum|direct decomposition]]
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| :<math>A=\bigoplus_{k=1}^m U_k</math>
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| into free <math>R</math>-submodules such that
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| # <math>i(U_k)=U_k</math>
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| # <math>J_k:=\bigoplus_{j=1}^k U_j</math> is a two-sided ideal of <math>A</math>
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| # <math>J_k/J_{k-1}</math> is a cell ideal of <math>A/J_{k-1}</math> w.r.t. to the induced involution.
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| Now <math>(A,i)</math> is called a cellular algebra if it has a cell chain. One can show that the two definitions are equivalent.<ref name="ccxi">{{citation|title= On the structure of cellular algebras|first1= S.|last1= König|first2= C.C.|last2= Xi|journal= Algebras and modules II. CMS Conference Proceedings|year=1996|pages=365–386}}</ref> Every basis gives rise to cell chains (one for each [[topological ordering]] of <math>\Lambda</math>) and choosing a basis of every left ideal <math>\Delta/J_{k-1}\subseteq J_k/J_{k-1}</math> one can construct a corresponding cell basis for <math>A</math>.
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| == Examples ==
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| === Polynomial examples ===
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| <math>R[x]/(x^n)</math> is cellular. A cell datum is given by <math>i=id</math> and
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| * <math>\Lambda:=\lbrace 0,\ldots,n-1\rbrace</math> with the reverse of the natural ordering.
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| * <math>M(\lambda):=\lbrace 1\rbrace</math>
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| * <math>C_{11}^\lambda := x^\lambda</math>
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| A cell-chain in the sense of the second, abstract definition is given by
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| : <math>0 \subseteq (x^{n-1}) \subseteq (x^{n-2}) \subseteq \ldots \subseteq (x^1) \subseteq (x^0)=R</math>
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| === Matrix examples ===
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| <math>R^{d\times d}</math> is cellular. A cell datum is given by <math>i(A)=A^T</math> and
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| * <math>\Lambda:=\lbrace 1 \rbrace</math>
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| * <math>M(1):=\lbrace 1,\dots,d\rbrace</math>
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| * For the basis one chooses <math>C_{st}^1 := E_{st}</math> the standard matrix units, i.e. <math>C_{st}^1</math> is the matrix with all entries equal to zero except the (''s'',''t'')-th entry which is equal to 1.
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| A cell-chain (and in fact the only cell chain) is given by
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| : <math> 0 \subseteq R^{d\times d}</math>
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| In some sense all cellular algebras "interpolate" between these two extremes by arranging matrix-algebra-like pieces according to the poset <math>\Lambda</math>.
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| === Further examples ===
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| Modulo minor technicalities all [[Hecke_algebra#Hecke_Algebras_of_Coxeter_Groups|Iwahori–Hecke algebras]] of finite type are cellular w.r.t. to the involution that maps the standard basis as <math>T_w\mapsto T_{w^{-1}}</math>.<ref>{{citation|title= Hecke algebras of finite type are cellular|first1= Meinolf|last1= Geck|journal= Inventiones Mathematicae|volume= 169|year= 2007|pages=501–517}}</ref> This includes for example the integral group algebra of the [[symmetric group]]s as well as all other finite [[Weyl group]]s.
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| A basic [[Brauer tree algebra]] over a field is cellular if and only if the Brauer tree is a straight line (with arbitrary number of exceptional vertices).<ref name="ccxi"/>
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| Further examples include q-[[Schur algebra]]s, the [[Brauer algebra]], the [[Temperley–Lieb algebra]], the [[Birman–Wenzl algebra|Birman–Murakami–Wenzl algebra]], the blocks of the Bernstein–Gelfand–Gelfand category <math>\mathcal{O}</math> of a [[semisimple Lie algebra]].<ref name="ccxi"/>
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| == Representations ==
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| === Cell modules and the invariant bilinear form ===
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| Assume <math>A</math> is cellular and <math>(\Lambda,i,M,C)</math> is a cell datum for <math>A</math>. Then one defines the ''cell module'' <math>W(\lambda)</math> as the free <math>R</math>-module with basis <math>\lbrace C_\mathfrak{s} | \mathfrak{s}\in M(\lambda)\rbrace</math> and multiplication
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| :<math>aC_\mathfrak{s} := \sum_{\mathfrak{u}} r_a(\mathfrak{u},\mathfrak{s}) C_\mathfrak{u}</math>
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| where the coefficients <math>r_a(\mathfrak{u},\mathfrak{s})</math> are the same as above. Then <math>W(\lambda)</math> becomes an <math>A</math>-left module.
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| These modules generalize the [[Specht module]]s for the symmetric group and the Hecke-algebras of type A.
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| There is a canonical bilinear form <math>\phi_\lambda: W(\lambda)\times W(\lambda)\to R</math> which satisfies
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| :<math>C_\mathfrak{st}^\lambda C_\mathfrak{uv}^\lambda \equiv \phi_\lambda(C_\mathfrak{t},C_\mathfrak{u}) C_\mathfrak{sv}^\lambda \mod A(<\lambda)</math>
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| for all indices <math>s,t,u,v\in M(\lambda)</math>.
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| One can check that <math>\phi_\lambda</math> is symmetric in the sense that
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| :<math>\phi_\lambda(x,y) = \phi_\lambda(y,x)</math>
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| for all <math>x,y\in W(\lambda)</math> and also <math>A</math>-invariant in the sense that
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| :<math>\phi_\lambda(i(a)x,y)=\phi_\lambda(x,ay)</math>
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| for all <math>a\in A</math>,<math>x,y\in W(\lambda)</math>.
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| === Simple modules ===
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| Assume for the rest of this section that the ring <math>R</math> is a field. With the information contained in the invariant bilinear forms one can easily list all simple <math>A</math>-modules:
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| Let <math>\Lambda_0:=\lbrace \lambda\in\Lambda | \phi_\lambda\neq 0\rbrace</math> and define <math>L(\lambda):=W(\lambda)/\operatorname{rad}(\phi_\lambda)</math> for all <math>\lambda\in\Lambda_0</math>. Then all <math>L(\lambda)</math> are [[Absolutely irreducible|absolute simple]] <math>A</math>-modules and every simple <math>A</math>-module is one of these.
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| These theorems appear already in the original paper by Graham and Lehrer.<ref name="grahamlehrer"/>
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| == Properties of cellular algebras ==
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| === Persistence properties ===
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| * Tensor products of finitely many cellular <math>R</math>-algebras are cellular.
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| * A <math>R</math>-algebra <math>A</math> is cellular if and only if its [[opposite algebra]] <math>A^{op}</math> is.
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| * If <math>A</math> is cellular with cell-datum <math>(\Lambda,i,M,C)</math> and <math>\Phi\subseteq\Lambda</math> is an [[Ideal (order theory)|ideal]] (a downward closed subset) of the poset <math>\Lambda</math> then <math>A(\Phi):=\sum RC_\mathfrak{st}^\lambda</math> (where the sum runs over <math>\lambda\in\Lambda</math> and <math>s,t\in M(\lambda)</math>) is an twosided, <math>i</math>-invariant ideal of <math>A</math> and the quotient <math>A/A(\Phi)</math> is cellular with cell datum <math>(\Lambda\setminus\Phi,i,M,C)</math> (where i denotes the induces involution and M,C denote the restricted mappings).
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| * If <math>A</math> is a cellular <math>R</math>-algebra and <math>R\to S</math> is a unitary homomorphism of commutative rings, then the [[extension of scalars]] <math>S\otimes_R A</math> is a cellular <math>S</math>-algebra.
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| * Direct products of finitely many cellular <math>R</math>-algebras are cellular.
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| If <math>R</math> is an [[integral domain]] then there is a converse to this last point:
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| * If <math>(A,i)</math> is a finite dimensional <math>R</math>-algebra with an involution and <math>A=A_1\oplus A_2</math> a decomposition in twosided, <math>i</math>-invariant ideals, then the following are equivalent:
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| # <math>(A,i)</math> is cellular.
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| # <math>(A_1,i)</math> and <math>(A_2,i)</math> are cellular.
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| * Since in particular all [[Block of a ring|blocks]] of <math>A</math> are <math>i</math>-invariant if <math>(A,i)</math> is cellular, an immediate corollary is that a finite dimensional <math>R</math>-algebra is cellular w.r.t. <math>i</math> if and only if all blocks are <math>i</math>-invariante and cellular w.r.t. <math>i</math>.
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| * [[Tits' deformation theorem]] for cellular algebras: Let <math>A</math> be a cellular <math>R</math>-algebra. Also let <math>R\to k</math> be a unitary homomorphism into a field <math>k</math> and <math>K:=Quot(R)</math> the [[quotient field]] of <math>R</math>. Then the following holds: If <math>kA</math> is semisimple, then <math>KA</math> is also semisimple.
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| If one further assumes <math>R</math> to be a [[local ring|local domain]], then additionally the following holds:
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| * If <math>A</math> is cellular w.r.t. <math>i</math> and <math>e\in A</math> is an [[Idempotent#Idempotent_ring_elements|idempotent]] such that <math>i(e)=e</math>, then the Algebra <math>eAe</math> is cellular.
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| === Other properties ===
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| Assuming that <math>R</math> is a field (though a lot of this can be generalized to arbitrary rings, [[integral domain]]s, [[local ring]]s or at least [[discrete valuation ring]]s) and <math>A</math> is cellular w.r.t. to the involution <math>i</math>. Then the following hold
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| * <math>A</math> is split, i.e. all simple modules are [[absolutely irreducible]].
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| * The following are equivalent:<ref name="grahamlehrer" />
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| # <math>A</math> is [[semisimple algebra|semisimple]].
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| # <math>A</math> is split semisimple.
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| # <math>\forall\lambda\in\Lambda: W(\lambda)</math> is simple.
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| # <math>\forall\lambda\in\Lambda: \phi_\lambda</math> is [[nondegenerate]].
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| * The [[Cartan matrix]] <math>C_A</math> of <math>A</math> is [[symmetric matrix|symmetric]] and [[positive-definite matrix|positive definite]].
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| * The following are equivalent:<ref>{{citation|title= Cellular algebras and quasi-hereditary algebras: A comparison|first1= S.|last1= König|first2= C.C.|last2= Xi|journal= [[Electronic Research Announcements of the American Mathematical Society]]|volume= 5|date= 1999-06-24|pages=71–75}}</ref>
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| # <math>A</math> is [[Quasi-hereditary algebra|quasi-hereditary]] (i.e. its module category is a [[highest-weight category]]).
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| # <math>\Lambda=\Lambda_0</math>.
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| # All cell chains of <math>(A,i)</math> have the same length.
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| # All cell chains of <math>(A,j)</math> have the same length where <math>j:A\to A</math> is an arbitrary involution w.r.t. which <math>A</math> is cellular.
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| # <math>\det(C_A)=1</math>.
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| * If <math>A</math> is [[Morita equivalent]] to <math>B</math> and the [[Characteristic (algebra)|characteristic]] of <math>R</math> is not two, then <math>B</math> is also cellular w.r.t. an suitable involution. In particular is <math>A</math> cellular (to some involution) if and only if its basic algebra is.<ref>{{citation|title= Cellular algebras: inflations and Morita equivalences|first1= S.|last1= König|first2= C.C.|last2= Xi|journal= Journal of the London Mathematical Society|volume= 60|year= 1999|pages=700–722}}</ref>
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| * Every idempotent <math>e\in A</math> is equivalent to <math>i(e)</math>, i.e. <math>Ae\cong Ai(e)</math>. If <math>char(R)\neq 2</math> then in fact every equivalence class contains an <math>i</math>-invariant idempotent.<ref name="ccxi"/>
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| ==References==
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| <references/>
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| [[Category:Algebras]]
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| [[Category:Representation theory]]
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