## Almost semisimple algebras

Last update: 11 May 2012

## Convolution algebras

### The decomposition theorem

Let $M$ be a smooth $G-$variety and let $N$ be a $G-$variety with finitely many $G-$orbits such that the orbit decomposition is an algebraic stratification of $N,$ $N = ⨆φGxφ, and μ:M→N$ is a $G-$equivariant projective morphism. Let ${𝒞}_{M}$ be the constant perverse sheaf on $M.$ The decomposition theorem [CG, 8.4.12] says that $μ*𝒞M = ⨁ i∈ℤ λ=(φ,χ)∈M^ L(λ,i) ⊗ ICλ[i] ≐ ⨁λ∈M^ L(λ) ⊗ ICλ, where L(λ) = ⨁i∈ℤ L(λ,i) ,$ ${\mu }_{*}$ is the derived functor of sheaf theoretic direct image, $\lambda$ runs over the indexes of the intersection cohomology complexes $I{C}^{\lambda },$ $L\left(\lambda \right)$ are finite dimensional vector spaces, and $&eqdot;$ indicates an equality up to shifts in the derived category.

### Convolution algebras

Let $\mu :M\to N$ be a proper map. The convolution algebra is $A = ExtDb(N)* ( μ*𝒞M, μ*𝒞M ) = ⨁k∈ℤ Extk ( μ*𝒞M, μ*𝒞M ).$ The decomposition theorem for ${\mu }_{*}{𝒞}_{M}$ induces a decomposition of $A.$ Since the intersection cohomology complexes $I{C}_{\phi }$ are the simple objects in the category of perverse sheaves, and the decomposition of $A$ simplifies to $A = ⨁λ∈M^ Endℂ(L(λ)) ⊕ ⨁k∈ℤ>0 ⨁λ,μ∈M^ Homℂ ( L(λ), L(μ) ) ⊗ ExtDb(N)k (ICλ,ICμ) .$ In the context there is a good theory of projective, standard and simple modules, and their decomposition matrices satisfy a BGG reciprocity. View elements of $A$ as sums $∑λ,μ ∑ P∈L^(λ), Q∈L^(μ) cPQλμ aPQλμ where cPQλμ ∈ℂ, and aPQλμ ∈ ⨁k>0 ExtDb(N)k (ICλ,ICμ).$ The algebra $A$ is completely controlled by the dimensions of the $L\left(\lambda \right)$ and the multiplication in $Abasic = Ext* (IC,IC) where IC = ⨁λ∈M^ ICλ,$ an algebra which has all one dimensional simple modules. The radical filtration of $A$ is $Radl(A) = ⨁λ,μ∈M^ Homℂ (L(λ),L(μ)) ⊗ ⨁k∈ℤ≥l ExtDb(N)k (ICλ,ICμ)$ and the nonzero

### Projective modules

Let ${e}^{\lambda }$ be a minimal idempotent in $\underset{\mu }{⨁}\mathrm{End}\left(L\left(\mu \right)\right).$ Then $P(λ) = Aeλ = L(λ) ⊕ ⨁ k>0 μ L(μ) ⊗ ExtDb(N)k (ICμ,ICλ)$ is the projective cover of the simple $A-$module $L\left(\lambda \right).$ Define an $A-$module filtration $P(λ) ⊇ P(λ)(1) ⊇ P(λ)(2) ⊇ ⋯$ by $P(λ)(m) = ⨁ k≥m μ L(μ) ⊗ ExtDb(N)k (ICμ,ICλ).$ Then Thus the multiplicity of the simple $A-$module $L\left(\mu \right)$ in a composition series of $P\left(\lambda \right)$ is $[P(λ) : L(μ)] = dim( Ext* ( IC𝕆,χ,IC𝕆′,χ′ ) ) = ∑k≥0 dim( ExtDb(N)k (ICμ,ICλ) ).$

### Standard and costandard modules

Let $\lambda =\left(\phi ,\chi \right),$ $x∈𝕆φ, and let ix: {x}↪N be the injection.$ Then ${i}_{x}^{!}{\mu }_{*}{𝒞}_{M}$ is the stalk of ${\mu }_{*}{𝒞}_{M}$ at $x$ and the Yoneda product makes $Δφ = H* (ix!𝒞M) = HomDb({x}) ( ℂ, ix!μ*𝒞M[*] ) = HomDb(N) ( (ix)!ℂ[-*], μ*𝒞M ), and, ∇φ = H* (ix*𝒞M) = H* ( {x}, ix*μ*𝒞M ) = HomDb({x}) ( ℂ, ix!μ*𝒞M[*] ) = HomDb(N) ( (ix)!ℂ[-*], μ*𝒞M ),$ into right $A-$modules. The action of an element $a\in {\mathrm{Ext}}^{k}\left({\mu }_{*}{𝒞}_{M},{\mu }_{*}{𝒞}_{M}\right)={\mathrm{Hom}}_{{D}^{b}\left(N\right)}\left({\mu }_{*}{𝒞}_{M},{\mu }_{*}{𝒞}_{M}\left[k\right]\right)$ sends $H* ( {x}, ix!μ*𝒞M ) → H*+k ( {x}, ix!μ*𝒞M ).$

A $G-$equivariant local system is a $G-$equivariant locally constant sheaf. The orbit ${𝕆}^{\phi }$ can be identified with $G/{G}_{x}$ where ${G}_{x}$ is the stabilizer of $x.$ ${\pi }_{0}\left({𝕆}^{\phi },x\right)={G}_{x}/{G}_{x}^{°}$ where ${G}_{x}^{°}$ is the connected component of the identity in ${G}_{x}.$ There is a homomorphism ${\pi }_{1}\left({𝕆}^{\phi },x\right)\to {\pi }_{0}\left({𝕆}^{\phi },x\right)={G}_{x}/{G}_{x}^{°}$ and the representations of ${\pi }_{1}\left({𝕆}^{\phi },x\right)$ on the fibers ${ℒ}_{x}$ of $G-$equivariant local systems $ℒ$ are exactly the pullbacks of finite dimensional representations of $C={G}_{x}/{G}_{x}^{°}$ to ${\pi }_{1}\left({𝕆}^{\phi },x\right).$ In this way the irreducible $G-$equivariant local systems on ${𝕆}^{\phi }$ can be indexed by (some of the) irreducible representations of ${G}_{x}/{G}_{x}^{°}$ [CG, Lemma 8.4.11]. There is an action of $C={G}_{x}/{G}_{x}^{°}$ on ${\Delta }^{\phi }$ which commutes with the action of $A.$ Similar arguments apply to ${\nabla }^{\phi }.$ As $\left(A,C\right)$ bimodules, $Δφ = ⨁χ∈C^ Δ(φ,χ)⊗χ and ∇φ = ⨁χ∈C^ ∇(φ,χ)⊗χ,$ and the standard and costandard $A-$modules are $Δ(λ) = Δ(φ,χ) and ∇(λ) = ∇(φ,χ).$ Using the decomposition theorem $Δ(λ) = H*( ix!𝒞M )χ = ⨁ k∈ℤ μ L(μ) ⊗ Hk( ix!ICμ )χ,$ where the subscript $\chi$ denotes the $\chi -$isotypic component. Define a filtration $Δ(λ) ⊇ Δ(λ)(1) ⊇ Δ(λ)(2) ⊇ ⋯ by Δ(λ)(m) = ⨁j≥m ⨁φ L(μ) ⊗ Hj(( ix!ICμ )χ.$ Then $\Delta {\left(\lambda \right)}^{\left(m\right)}$ is an $A-$module and $\mathrm{gr}\left(\Delta \left(\lambda \right)\right)$ is a semisimple $A-$module. This (and a similar argument for $\nabla \left(\lambda \right)$) show that the multiplicity of the simple $A-$module $L\left(\mu \right)$ in composition series of $\Delta \left(\lambda \right)$ and $\nabla \left(\lambda \right)$ are $[Δ(λ):L(μ)] = ∑kdim( Hk( ix!ICμ )χ ) and [∇(λ):L(μ)] = ∑kdim( Hk( ix*ICμ )χ ).$ Define the standard KL-polynomial and the costandard KL-polynomial of $A$ to be $PλμΔ(t) = ∑ktk dim( Hk( ix!ICμ )χ ) and Pλμ∇(t) = ∑ktk dim( Hk( ix*ICμ )χ ),$ respectively. Then ??? says that $[Δ(λ):L(μ)] = PλμΔ(1) and [∇(λ):L(μ)] = Pλμ*(1).$ These identities are analogues of the original Kazhdan-Lusztig conjecture describing the multiplicities of simple $𝔤-$modules in Verma modules.

### The contravariant form

Note that there is a canonical homomorphism $Δ(λ) →cλ ∇(λ)$ coming from applying the functor ${H}^{*}$ to the composition $(ix)! (ix)! μ*𝒞M → μ*𝒞M → (ix)* (ix)* μ*𝒞M,$ where the two maps arise from the canonical adjoint functor maps. Use the map ${c}_{\lambda }$ to define a bilinear form on $\Delta \left(\lambda \right)$ by $⟨,⟩: Δ(λ) ⊗ Δ(λ) → ℂ m1⊗m2 ↦ m1∩cλ(m2)$ Then $L(λ) = Δ(λ) / Rad(⟨,⟩).$

There is an involutive automorphism ${\phantom{A}}^{t}:A\to A$ on $A$ (coming from switching the two factors in $Z=M{×}_{N}M$). If $M$ is an $A-$module the contragradient module is Then $∇(λ) ≅ Δ(λ)*.$

### Reciprocity

If $\lambda =\left(\phi ,\rho \right)$ define Then $[P(λ):L(μ)] = ∑kdim ExtDb(N)k (ICλ,ICμ) = ∑kdim Ext Db(N) dλ+dμ+k (ICλ,ICμ) = ∑k (-1)k dimExt Db(N) dλ+dφ+k (ICλ,ICμ) = (-1)dφ+dψ ∑𝕆χ( 𝕆, i𝕆! ICφ∨ ⊗! i𝕆! ICψ ) = (-1)dφ+dψ ∑𝕆χ 𝕆, (-1)dφ ∑α,k [ ℋk i𝕆! (ICφ∨) : α ] α ⊗! (-1)dψ ∑β,l [ ℋl i𝕆! (ICψ) : β ]β = ∑𝕆,α,β χ 𝕆, ∑k [ ℋk i𝕆! (ICφ) : α* ]α ⊗! ∑l [ ℋl i𝕆! (ICψ) : β ]β = ∑α,β∑k dimℋk (iα!ICφ) ∑𝕆 χ( 𝕆, α*⊗!β ) ∑l dimℋl (iβ!ICψ) = ∑α,β [ℳα!:Lφ] ∑𝕆 χ( 𝕆, α*⊗β ) [ℳβ!:Lψ] = ∑α,β Pφα(1) Dαβ Pψβ(1) = (PDPt)φψ,$ where

1. the third equality follows from the vanishing of Ext groups in odd degrees,
2. $\chi$ denotes the Euler characteristic,
3. $P$ is the matrix $\left({P}_{\phi \alpha }\left(1\right),$ and
4. $D$ is the matrix $\left(\sum _{𝕆}\chi \left(𝕆,{\alpha }^{*}\otimes \beta \right)\right).$
This identity is the "BGG reciprocity" for the algebra $A.$

### The Steinberg variety

Let $x\in N$ and define $Z = M×NM = { (m1,m2)∈M×M | μ(m1) = μ(m2) } and Mx = μ-1(x).$ There are commutative diagrams $Z=M×NM M×M N=NΔ N×N ι Δ μ12 μ1×μ2 and Mx M {x} N ι ix μ μ$ which (via base change) provide isomorphisms $H*(Z) = HomDb(Z12) ( ℂZ12, ( ℂZ12[*] )∨ ) = HomDb(Z12) ( μ12*ℂN, ι! 𝒞M1×M2 [m1+m2][-*] ) = HomDb(N) ( ℂN, (μ12)*ι! 𝒞M1×M2 [m1+m2-*] ) = HomDb(N)( ℂN, Δ! (μ1×μ2)* ( 𝒞M1 ⊠ 𝒞M2 ) [m1+m2-*] ) = HomDb(N) ( ℂN, Δ!( (μ1)*𝒞M1 ⊠ (μ2)*𝒞M2 ) [m1+m2-*] ) = Ext Db(N) m1+m2-* ( (μ1)* 𝒞M1, (μ2)* 𝒞M2 ), H*(Mx) = HomDb(Mx) ( ℂMx, ( ℂMx [*] )∨ ) = HomDb(Mx) ( μ*ℂ{x}, ( (ι*ℂM)[*] )∨ ) = HomDb({x}) ( ℂ{x}, μ*( ι!ℂM[2m] ) [-*] ) = HomDb({x}) ( ℂ{x}, ix!μ*𝒞M [m-*] ) = Hm-* ( ix!μ*𝒞M ),$ and $H*(Mx) = HomDb(Mx) ( ℂMx, ℂMx[*] ) = HomDb(Mx) ( μ*ℂ{x}, ℂMx[*] ) = HomDb({x}) ( ℂ{x}, μ*ℂMx[*] ) = HomDb({x}) ( ℂ{x}, μ!ι* ℂM[*] ) = HomDb({x}) ( ℂ{x}, ix*μ!ℂM[*] ) = HomDb({x}) ( ℂ{x}, ix*μ* 𝒞M[*-m] ) = H*-m ( ix*μ*𝒞M ).$

### The category ${D}^{b}\left(N\right)$

The category ${\mathrm{Comp}}^{b}\left(\mathrm{Sh}\left(N\right)\right)$ is the category of all finite complexes $A = ( 0→A-m →A-m+1 →⋯→ An-1→ An→0 ), m,n∈ℤ≫0,$ of sheaves on $N$ with morphisms being morphisms of complexes which commute with the differentials. The ${j}^{\mathrm{th}}$ cohomology sheaf of $A$ is $ℋj(A) = ker(Aj→Aj+1) im(Aj-1→Aj) .$ A morphism in ${\mathrm{Comp}}^{b}\left(\mathrm{Sh}\left(N\right)\right)$ is a quasi-isomorphism if it induces isomorphisms on cohomology. The category ${D}^{b}\left(\mathrm{Sh}\left(N\right)\right)$ is the category ${\mathrm{Comp}}^{b}\left(\mathrm{Sh}\left(N\right)\right)$ with additional morhpisms obtained by formally inverting all quasi-isomorphisms.

Assume that $N$ is a $G-$variety with a finite number of orbits such that the $G-$orbit decomposition A constructible sheaf is a sheaf that is locally constant on strata of $N.$ A constructible complex is a complex such that all of its cohomology sheaves are constructible.

The derived category of bounded constructible complexes of sheaves on $N$ is the full subcategory ${D}^{b}\left(N\right)$ of ${D}^{b}\left(\mathrm{Sh}\left(N\right)\right)$ consisting of constructible complexes. Full means that the morphisms in ${D}^{b}\left(N\right)$ are the same as those in ${D}^{b}\left(\mathrm{Sh}\left(N\right)\right).$

The shift functor $\left[i\right]:{D}^{b}\left(N\right)\to {D}^{b}\left(N\right)$ is the functor that shifts all complexes by $i.$

The Verdier duality functor ${\phantom{A}}^{\vee }:{D}^{b}\left(N\right)\to {D}^{b}\left(N\right)$ is defined by requiring where $\Delta :N\to N×N$ is the diagonal map.

The Verdier duality functor satisfies the properties $(A∨)∨ = A, (A[i])∨ = A∨ [-i], and HomDb(N) (A1,A2) = HomDb(N) (A2∨,A1∨).$ Define respectively. The Yoneda product $ExtDb(N)p (A1,A2) × ExtDb(N)q (A2,A3) → ExtDb(N)p+q (A1,A3)$ is given by $HomDb(N) (A1,A2[p]) × HomDb(N) ( A2[p], A3[p+q] ) → HomDb(N) (A1, A3[p+q]),$ using the canonical identification ${\mathrm{Hom}}_{{D}^{b}\left(N\right)}\left({A}_{2},{A}_{3}\left[q\right]\right)\cong {\mathrm{Hom}}_{{D}^{b}\left(N\right)}\left({A}_{2}\left[p\right],{A}_{3}\left[p+q\right]\right).$

If $f:X\to Y$ is a morphism define $f* = derived functor of sheaf theoretic direct image, f* = derived functor of sheaf theoretic inverse image,$ Then $HomDb(X) ( f*A1, A2 ) = HomDb(Y) ( A1, f*A2 ), and, HomDb(X) ( A2, f!A1 ) = HomDb(Y) ( f!A2, A1 ).$ If $f:X\to Z$ and $g:Y\to Z$ define the base change formula as where $X{×}_{Z}Y=\left\{\left(x,y\right)\in X×Y\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}f\left(x\right)=g\left(y\right)\right\}.$

The category of perverse sheaves on $X$ is a full subcategory of ${D}^{b}\left(X\right)$ which is abelian. The simple objects in the category of perverse sheaves are the intersection cohomology complexes $ICφ indexed by pairs φ = (𝕆,χ),$ where $𝕆$ is a $G-$orbit on $X$ and $\chi$ is an irreducible local system on $X.$ By ???, the local systems $\chi$ on $𝕆$ can be identified with (some of the) representations of the component group ${Z}_{G}\left(x\right)/{Z}_{G}{\left(x\right)}^{°}$ where $x$ is a point in $𝕆.$ If $X$ is smooth the constant perverse sheaf ${𝒞}_{X}$ on $X$ is given by $|𝒞X|Xi = ℂXi [dimℂXi],$ on the irreducible components of $X.$ Since the intersection cohomology complexes $I{C}_{\phi }$ are the simple objects in the category of perverse sheaves,

## Dlab-Ringel algebras

Let $C$ and $D$ be rings, a $\left(C,C\right)$ bimodule homomorphism. Define an algebra $A = C⊕D⊕L⊕ R⊕R⊗CL$ and product determined by the multiplication in $C$ and $D,$ the module structure of $R$ and $L$ and the additional relations $cr=0, dl=0, rd=0, lc=0, and (r1⊗l1)(r2⊗l2) = r1⊗ε(l1⊗r2)l2.$ Let

1. ${e}_{C}$ be the image of the identity of $C$ in $A,$ and
2. ${e}_{D}$ be the image of the identity of $D$ in $A.$
Then, if $e={e}_{C}$ then $1=eC+eD, C = eCAeC, L = eCAeD, R = eDAeC, D′ = eDAeD,$ so that $A = { | c l r d′ | c∈C, l∈L, r∈R, d′∈D′ }$ with matrix multiplication. Then
1. ${e}_{D}A{e}_{D}=D+R{\otimes }_{C}L$ is a subring of $A,$ and
2. $R{\otimes }_{C}L$ is an ideal in ${e}_{D}A{e}_{D},$ and
3. $R{\otimes }_{C}L={e}_{D}A{e}_{C}A{e}_{D}.$

### Structure of $Z\left(\epsilon \right)$

Let The left radical $L\left(\epsilon \right)$ and the right radical $R\left(\epsilon \right)$ of $\epsilon$ are defined by $L(ε) = {l∈L | ε(l⊗r)∈Rad(C), for all r∈R}, R(ε) = {r∈R | ε(l⊗r)∈Rad(C), for all l∈L}.$ The map $\epsilon$ is nondegenerate if and $R\left(\epsilon \right)=0.$ Let $C_ = C/Rad(C), L_ = L/L(ε), R_ = R/R(ε), and φ: R⊗CL → R_⊗C_L_ r_⊗l_ ↦ r⊗l_.$ Then $\mathrm{ker}\phi$ is generated by $R{\otimes }_{C}L\left(\epsilon \right)$ and $R\left(\epsilon \right){\otimes }_{C}L,$ and we have that $\mathrm{ker}\phi \cdot R\subseteq R\left(\epsilon \right)$ and $L\cdot \mathrm{ker}\phi \subseteq L\left(\epsilon \right).$ Then and $A(ε)I ≅ A(ε_) where the map ε_: L_⊗DR_ → C_ l⊗r ↦ l_⊗r_$ is a nondegenerate $\left(\stackrel{_}{C},\stackrel{_}{C}\right)$ bimodule homomorphism.

If $\epsilon :L{\otimes }_{D}R\to C$ is nondegenerate and $R$ is a projective $C-$module then there is a $\left(D,C\right)$ bimodule isomorphism $τ: R →∼ L* r ↦ λr: L → C l ↦ ε(l⊗r) so that ε = ev∘(id⊗τ)$ and $A(ε) ≅ A(evL).$

If $C,D,L,R$ are finite dimensional vector spaces over $𝔽$ and $D=𝔽$ then $ε = ε0⊕evP: ( L0⊕P* ) ⊗D ( R0⊕P ) → C,$ with $P$ projective and $\mathrm{im}{\epsilon }_{0}\subseteq \mathrm{Rad}\left(C\right).$

If $\epsilon ={\epsilon }_{0}\oplus {\mathrm{ev}}_{P}$ with $P$ finitely generated and projective then $A(ε)-mod →∼ A(ε0)-mod M ↦ eM where e = 1-∑i pi⊗αi.$

If $\mathrm{im}\epsilon \subseteq \mathrm{Rad}\left(C\right)$ then $Rad(A(ε0)) = I = Rad(C) ⊕ Rad(D) ⊕ L0 ⊕ R0 ⊕ R0⊗CL0$ and $A(ε0) Rad(A(ε0)) ≅ C Rad(C) ⊕ D Rad(D) .$

### The module category of $Z\left(\epsilon \right)$

Let $𝒞$ and $𝒟$ be categories $F:𝒞→𝒟 and G:𝒞→𝒟 be functors, and F→εG,$ a natural transformation. Define a category $𝒜$ with

1. Objects:
2. Morphisms: $\left(f,g\right)$ with such that $FM GM V εM m n FM′ GM′ V′ εM′ m′ n′ commutes.$
A fundamental case is when $𝒟$ is the category of vector spaces over $𝔽.$

The equivalence between the category $𝒜$ and the module category of $Z\left(\epsilon \right)$ is given by letting $𝒞=C-mod$ and $𝒟=D-mod$ and $F: 𝒞 → 𝒟 M ↦ R⊗CM and G: 𝒞 → 𝒟 M ↦ HomC(L,M)$ where the $D-$action on ${\mathrm{Hom}}_{C}\left(L,M\right)$ is given by Then let $\epsilon :F\to G$ be the natural transformation given by $ε: F → G R⊗CM →εM HomC(L,M) r⊗m ↦ τ: L → M l ↦ ε(l⊗r)m.$ Then $𝒜 →∼ A-mod (X,Y,ρ,λ) ↔ M where X=eM, Y=(1-e)M,$ and the $L-$action and $R-$action on $M$ define $\rho$ and $\lambda$ via Note that and $R⊗CX = FX GX Y εX ρ λ = HomC(L,X)$ commutes.

### Macpherson-Vilonen

Let $X$ be a Thom-Mather stratified space with a fixed stratification such that all strata have even codimension. Let Let $S$ be a closed stratum such that $S$ is contractible and let $ι:X-S ↪X,$ be the inclusion. Let Let $F: P(X-S) → {vector spaces} P ↦ ℍ-d-1 (K;P)$ and $G: P(X-S) → {vector spaces} P ↦ ℍ-d (L,K;P) = ℍ-d (L,j! P|L-K).$ Let $𝒜$ be the corresponding category as in the previous section. Then the map $P(X) →∼ 𝒜 Q ↦ |Q|X-S ℍ-d-1 (K,Q) ℍ-d (L,K;Q) ℍ-d (𝔻,K;Q) εX ρ λ$ is an equivalence of categories, where ${\phantom{|}Q|}_{X-S}={\iota }^{*}Q,$ and ${\epsilon }_{Q}$ is the coboundary homomorphism in the long exact sequence for the pair $L,K.$ What is $𝔻$????

Examples.

1. The flag variety.
2. The nilpotent cone.

## Quasihereditary algebras

Let $𝔽$ be a field. A separable algebra over $𝔽$ is an algebra $A$ such that $ARad(A) ≅ ⨁λ ϵA^ Mdλ (𝔽).$ Two algebras $A$ and $B$ are Morita equivalent if $\mathrm{Mod}-A$ is equivalent to $\mathrm{Mod}-B$ (Check this in Gelfand-Manin).

A ring $A$ is semiprimary if there is a nilpotent ideal $\mathrm{Rad}\left(A\right)$ such that $A/\mathrm{Rad}\left(A\right)$ is semisimple artinian. Note: If $A$ is finite dimensional then $A$ is semiprimary.

A hereditary ring is a ring $A$ such that every submodule of a projective module is projective.

A hereditary ideal is an ideal $J$ such that

1. $J$ is projective as a right $A-$module,
2. ${J}^{2}=J,$ and
3. $J\mathrm{Rad}\left(A\right)J=0.$
Note: ${J}^{2}=J$ if and only if there is an idempotent $e\in A$ with $J=AeA.$

A quasihereditary ring is a semiprimary ring $A$ with a chain of ideals for each $1\le l\le m-1.$

Let $A$ be a quasihereditary algebra $0=J0 ⊆J1 ⊆⋯⊆ Jm=A.$ Let $e$ be an idempotent in $A$ such that $Jm-1 = AeA and eA(1-e) ⊆ Rad(A).$ Let $C=eAe and D = AAeA = AJm-1$ and $L D C = eA(1-e) and R D C = (1-e) Ae$ and let $ε: L⊗DR → C l⊗r ↦ lr.$ Assume $D$ is a separable $k-$algebra. Then

1. $D+\left(1-e\right)AeA\left(1-e\right)=\left(1-e\right)A\left(1-e\right),$
2. $A=C\left(\epsilon \right),$
3. $C$ is quasihereditary with heredity chain $0=I0 ⊆⋯⊆ Im-1 =C, where Il = eJle.$

### Highest weight categories

Let $A$ be a finite dimensional algebra and let $\stackrel{^}{A}$ be an index set for Then $𝒜=A-mod$ is a highest weight category if $P\left(\lambda \right)$ has a filtration $0= P(λ)(m) ⊆⋯⊆ P(λ)(1) ⊆P(λ),$ with for $1\le k\le m-1.$

Highest weight categories satisfy BGG-reciprocity, $[I(λ):∇(μ)] = [Δ(μ):L(λ)].$

 Proof. Since if follows that $Hom(Δ(λ),M) = ( number of ∇(λ) in a ∇-filtration of M ).$ Thus $[I(μ) : ∇(λ)] = dim(Hom(Δ(λ),I(μ))) dim(End(L(λ))) = [Δ(λ) : L(μ)].$ How does this proof compare to the proof for convolution algebras in Chriss and Ginzburg? $\square$

Examples of highest weight categories.

1. the category of finite dimensional rational $G-$modules, and $\nabla \left(\lambda \right)={H}^{0}\left(G/B,{ℒ}_{\lambda }\right),$
2. $𝒜$ the category $𝒪,$ and $\nabla \left(\lambda \right)=M{\left(\lambda \right)}^{\vee }.$
Vogan, Irreducible characters of semisimple Lie groups II; The Kazhdan-Lusztig conjectures

Let $A$ be a finite dimensional algebra and let $𝒜=A-$mod. Then $𝒜$ is a highest weight category if and only if $A$ is a quasihereditary algebra.

 Proof. ⇒) Assume $𝒜$ is a highest weight category. Let $\lambda$ be a maximal weight and let $P(λ) = Aeλ and JAeλA.$ Then $J$ is projective as a left $A-$module, $HomA(J,A/J) = 0, J⋅Rad(J) = 0.$ So $J$ is a hereditary ideal. Finally, $\left(A/J\right)-$mod is a highest weight category with $\stackrel{^}{\left(A/J\right)}=\stackrel{^}{A}-\left\{\lambda \right\}.$ ⇐) Assume $A$ is a quasihereditary algebra, $0=J0⊆J1 ⊆⋯⊆ Jm=A.$ Define $\lambda <\mu$ if with $i Suppose $i$ is (the unique integer) such that $L\left(\lambda \right)$ appears in $\left({J}_{i}/{J}_{i-1}\right)/\mathrm{Rad}\left({J}_{i}/{J}_{i-1}\right)$ and let Then $L\left(\lambda \right)$ is the simple head of $A\left(\lambda \right)$ and, since ${J}_{i-1}\cdot \mathrm{Rad}\left(A/{J}_{i-1}\right)\cdot {J}_{i-1}=0,$ all other composition factors of $A\left(\lambda \right)$ are lower. If $L\left(\lambda \right)$ is a simple $A-$module then there is an idempotent ${e}_{\lambda }\in A$ such that $P\left(\lambda \right)=A{e}_{\lambda }$ (${e}_{\lambda }$ is a minimal idempotent). Then $0=J0eλ ⊆J1eλ ⊆⋯⊆ Jmeλ = Aeλ = P(λ)$ is a good filtration of $P\left(\lambda \right).$ $\square$

### Duals and Projectives

Let $L$ be a $C-$module and let $Z=EndC(L)$ so that $L$ is a $\left(C,Z\right)$ bimodule. The dual module to $L$ is the $\left(Z,C\right)$ bimodule $L* = HomC(L,C).$ The evaluation map is the $\left(C,C\right)$ bimodule homomorphism $ev: L⊗ZL* → C l⊗λ ↦ λ(l)$ and the centralizer map is the $\left(Z,Z\right)$ bimodule homomorphism $ξ: L*⊗CL → Z λ⊗l ↦ zλ,l : L → L m ↦ λ(m)l.$ Recall that [Bou, Alg. II §4.2 Cor.]

1. $L$ is a projective $C-$module if and only if $1\in \mathrm{im}\xi ,$
2. If $L$ is a projective $C-$module then $\xi$ is injective,
3. If $L$ is a finitely generated projective $C-$module then $\xi$ is bijective,
4. If $L$ is a finitely generated free module then $ξ-1(z) = ∑i bi*⊗z(bi),$ where $\left\{{b}_{1},...,{b}_{d}\right\}$ is a basis of $L$ and $\left\{{b}_{1}^{*},...,{b}_{d}^{*}\right\}$ is the dual basis in ${M}^{*}.$
Statement (a) says that $L$ is projective if and only if there exist ${b}_{i}\in L$ and ${b}_{i}^{*}\in {L}^{*}$ such that

## Cellular algebras

A cellular algebra is an algebra $A$ with $a basis {| aSTλ| λ∈A^, S,T∈A^λ} an involutive antihomomorphism A*: A→A, and a partial order ≤ on A^$ such that

1. ${\left({a}_{ST}^{\lambda }\right)}^{*}={a}_{TS}^{\lambda },$
2. If $A\left(<\lambda \right)=\mathrm{span-}\left\{\phantom{|}{a}_{ST}^{\mu }|\mu <\lambda \right\}$ then $aaSTλ = ∑Q∈A^λ Aλ(a)QT aQTλ modA(<λ), for all a∈A.$
Applying the involution ${\phantom{A}}^{*}$ to (b) and using (a) gives that $aTSλ a* = ∑Q∈A^λ Aλ(a)QS aTQλ modA(<λ), for all a∈A.$

The concept of a cellular algebra is not really the "right" one. The "right" one comes from the structure of a convolution algebra whenever the decomposition theorem holds [CG, 8.6.9].

## Peter Webb's generalized reciprocity

Let $𝔬$ be a complete discrete valuation ring, $k=𝔬/𝔭$ its residue field and let ${}_{𝔬}A$ be an algebra over $𝔬,$ $k ← 𝔬 → 𝕂 Ak ← A𝔬 → A𝕂$

The diagram $K0 (A𝕂 ) G0 (A𝕂 ) K0 (Ak ) G0 (Ak ) cA cλ e=Dt D$ commutes, where $e$ is defined by lifting idempotents. Furthermore $e={D}^{t}.$

 Proof. If $P$ is projective, $U$ any finitely generated module, put $⟨P,U⟩ = dimHom(P,U).$ This is well defined on ${K}_{0}\left({}_{𝕂}A\right)×{G}_{0}\left({}_{𝕂}A\right)$ and ${K}_{0}\left({}_{k}A\right)×{G}_{0}\left({}_{k}A\right).$ Then $e(P) = 𝕂⊗𝔬P^, where k⊗𝔬P^ = P.$ $\square$

Let ${U}_{0}$ be a $𝔬-$form of $U$ and let $P$ be projective. Then ${\mathrm{Hom}}_{{}_{𝔬}A}\left(\stackrel{^}{P},{U}_{0}\right)$ is an $𝔬-$lattice in ${\mathrm{Hom}}_{{}_{𝕂}A}\left(K{\otimes }_{𝔬}\stackrel{^}{P},U\right)$ and the morphism ${\mathrm{Hom}}_{{}_{𝔬}A}\left(\stackrel{^}{P},{U}_{0}\right)\to {\mathrm{Hom}}_{{}_{k}A}\left(P,{U}_{0}/𝔭{U}_{0}\right)$ is reduction mod $𝔭.$

$\mathrm{dim}{\mathrm{Hom}}_{{}_{𝕂}A}\left(K{\otimes }_{𝔬}\stackrel{^}{P},U\right)={\mathrm{rank}}_{𝔬}{\mathrm{Hom}}_{{}_{𝔬}A}\left(\stackrel{^}{P},{U}_{0}\right)=\mathrm{dim}{\mathrm{Hom}}_{{}_{k}A}\left(P,{U}_{0}/𝔭{U}_{0}\right).$

This shows that $e$ and $D$ are the transpose of each other with respect to the forms. The diagram commutes from the definition of $e.$

The Cartan matrix $CAk = DCA𝕂 Dt$ where ${C}_{{}_{𝕂}A}$ is the Cartan matrix of $A.$

If ${}_{𝕂}A$ is semisimple then ${C}_{{}_{𝕂}A}=\mathrm{id}.$

## The category $𝒪$

Let $U$ be a $ℤ$ graded algebra with

1. ${U}_{0}$ reductive,
2. ${U}_{i}$ finite dimensional,
3. $U$ semisimple under the adjoint action.
The category $𝒪$ is the category of $ℤ$ graded $U$ modules which are
1. ${U}_{0}$ semisimple, and
2. ${U}_{\ge 0}$ locally finite.
Define $𝒪≤n = {M∈𝒪 | Mi=0 if i>n}.$

### Standard and costandard modules

Let ${\stackrel{^}{U}}_{0}$ be an index set for the finite dimensional $ℤ-$graded ${U}_{0}$ modules. The Verma module or standard module and the coVerma module or costandard module are given by Let $M\in 𝒪.$ A $\Delta -$flag for $M$ is an increasing filtration $0=M(0) ⊆M(1) ⊆M(2) ⊆⋯ such that M = ⋃iM(i),$ and, for each $i\ge 1,$ ${M}^{\left(i\right)}/{M}^{\left(i-1\right)}\cong \Delta \left({\lambda }^{\left(i\right)}\right)$ for some ${\lambda }^{\left(i\right)}\in {\stackrel{^}{U}}_{0}.$

1. $\Delta \left(\lambda \right)$ has simple head $L\left(\lambda \right).$
2. $\nabla \left(\lambda \right)$ has simple socle $L\left(\lambda \right).$
3. $\left\{L\left(\lambda \right)\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}\lambda \in {\stackrel{^}{U}}_{0}\right\}$ are the simple objects in $𝒪.$

1. $\Delta \left(\lambda \right)$ is the projective cover of $L\left(\lambda \right)$ in ${𝒪}_{\le |\lambda |}.$
2. $\nabla \left(\lambda \right)$ is the injective hull of $L\left(\lambda \right)$ in ${𝒪}_{\le |\lambda |}.$
3. ${\mathrm{Hom}}_{𝒪}\left(\Delta \left(\mu \right),\nabla \left(\lambda \right)\right)=\left\{\begin{array}{ll}0,& if\phantom{\rule{.5em}{0ex}}\lambda \ne \mu ,\\ ℂ,& if\phantom{\rule{.5em}{0ex}}\lambda =\mu .\end{array}\phantom{\right\}}$
4. ${\mathrm{Ext}}_{𝒪}^{1}\left(\Delta \left(\mu \right),\nabla \left(\lambda \right)\right)=0.$

### Projectives

If $K=⨁{K}_{i}$ is a $ℤ$ graded ${U}_{\ge 0}$ module define $τ≥n = K ⨁i>nKi = ⨁i≤nKi.$ If $\lambda \in {\stackrel{^}{U}}_{0}$ define $Q = U⊗U≥0τ≤n( U≥0⊗U0U0λ ),$ and let ${P}_{\le n}\left(\lambda \right)$ be an indecomposable summand of $Q$ which has $L\left(\lambda \right)$ as a quotient and define ${K}_{m,n},$ for $m\ge n$ by the exact sequence $0→Km,n →P≤m(λ) →P≤n(λ) →0.$

1. $Q$ is projective and $Q\to L\left(\lambda \right)\to 0.$
2. ${P}_{\le n}\left(\lambda \right)$ is a projective cover of $L\left(\lambda \right)$ in ${𝒪}_{\le n}.$
3. ${P}_{\le n}\left(\lambda \right)$ has a $\Delta$ flag.
4. ${K}_{m,n}$ has a $\Delta$ flag.
5. $L\left(\lambda \right)$ has a projective cover in $P\left(\lambda \right)$ in $𝒪$ if and only if the projective system ${P}_{\le m}\left(\lambda \right)\to {P}_{\le n}\left(\lambda \right)$ stabilizes, in which case

### Tilting modules

Let $\lambda \in {\stackrel{^}{U}}_{0}.$ A tilting module is a module that has both a $\Delta$ flag and a $\nabla$ flag.

There is a unique indecomposable tilting module $T\left(\lambda \right)$ of highest weight $\lambda .$

### Blocks

Define $\ge$ on ${\stackrel{^}{U}}_{0}$ by $μ≥λ if [Δ(μ):L(λ)] ≠ 0 or [∇(μ):L(λ)] ≠ 0.$ Let $\left[\lambda \right]$ denote the equivalence class of $\lambda$ with respect to the equivalence relation generated by $\ge .$ Define $𝒪[λ] = {M∈𝒪 | if [M:L(μ)]≠0 then μ∈[λ]},$ and for $M\in 𝒪$ define $M[λ] = U( ∑im( P≤n(λ) →φ M ) ),$ the submodule of $M$ generated by the images of morphisms $\phi :{P}_{\le n}\left(\lambda \right)\to M.$

### Multiplicities

Let $𝒜$ be an abelian category and let $L$ be simple. Let $m\in 𝒜.$ The multiplicity of $L$ in $M$ is $[M:L] = supF Card {i | FiM/Fi+1M ≅ L},$ where the supremum is over all (finite) filtrations of $M.$

If $M\in {𝒪}_{\le n}$ and $N\in 𝒪$ with a $\Delta$-flag then $[M:L(λ)] = dimHom𝒪 (P≤n(λ),M) and [N:Δ(μ)] = dimHom(N,∇(μ)).$ Thus $[P≤n(λ):Δ(μ)] = [∇(μ):L(λ)], for λ,μ∈U^0 and n≥max{ |λ|, |μ| }.$

## The category ${𝒪}_{\mathrm{int}}$

Start with $U={U}_{<0}{U}_{0}{U}_{>0}.$ $𝒪int = { M∈U-mod | M∈U0ss, M∈U>0nilp, M∈U<0nilp }.$

## Finite dimensional algebras

Let $A$ be a finite dimensional algebra.

1. The projective indecomposables are $Ae$ for a minimal idempotent $e$ of $A.$
2. The simples $L\left(\lambda \right)$ are the simple heads of the projective indecomposables $P\left(\lambda \right).$
3. The blocks are $Az$ for a minimal central idempotent $z$ of $A.$
4. The Cartan matrix is $[P(λ) : L(μ)].$

## Temperley-Lieb algebras

### Computation of the ${\epsilon }_{\sigma }^{\gamma }$

The quantum dimensions of the finite dimensional simple ${U}_{q}{\mathrm{𝔰𝔩}}_{2}$ modules are $dimq( L(k-2j) ) = ∏b∈(k-j) [2+c(b)] [h(b)] = ∏i=0k-j-1 [2+i] [k-j-i] = [k-j+1] = [dim(L(k-2j))].$ As a $\left({U}_{q}{\mathrm{𝔰𝔩}}_{2},T{L}_{k}\left(n\right)\right)$ bimodule $V⊗k ≅ ∑j=0⌊k2⌋ L(k-2j) ⊗ TLk(k-j,j).$ Thus and $trq( aZXσ ) = δZX dimq(L(σ)) and trq( b ZX σμγ ) = δ ZX σμ dimq(L(γ)).$ If $a\in A$ then $trq (aek) = trq(a) trq(ek) = ntrq(a), and so trq( ε1(b) ) = 1n trq( ε1(b)ek ) = 1n trq( ekbek ) = 1n trq( bek2 ) = trq(bek) = trq( b(Tk-q) ) = (z-q) trq(b) = ( q2n-q ) trq(b) = 1n trq(b).$ So $1n dimq(L(γ)) = 1n trq( b ZX σμ γ ) = trq( ε1 ( b ZX σμ γ ) ) = trq( εσγ a ZX σ ) = εσγ dimq(L(σ)).$ Thus $εσγ = [dim(L(γ))] n⋅[dim(Lσ))] .$

### Generators and relations

The Temperley-Lieb algebra, $ℂ{T}_{k}\left(n\right),$ is the algebra over $ℂ$ given by generators ${E}_{1},{E}_{2},...,{E}_{k-1}$ and relations $EiEj = EjEi, if |i-j|>1, Ei Ei±1 Ei = Ei, and Ei2 = nEi.$

If $[2] = q+q-1 = n then q = 12 ( n+ n2-4 ), q-1 = 12 ( n- n2-4 ),$ since ${q}^{2}-nq+1=0.$ Then $[k] = qk-q-k q-q-1 = 12k-1 ∑m=1 k+12 ( k 2m-1 ) nk-2m+1 (n2-4)m-1.$ The problem with this expression is that it is not clear that $\left[k\right]$ is a polynomial in $n$ with integer coefficients (which alternate in sign?).

The Iwahori-Hecke algebra ${H}_{k}\left(q\right)$ is the algebra over $ℂ$ with generators ${T}_{1},{T}_{2},...,{T}_{k-1}$ and relations $TiTj = TjTi, if |i-j| >1, TiTi±1Ti = Ti+1TiTi+1, if 2≤i≤k-1, Ti2 = (q-q-1)Ti+1.$ There is a surjective algebra homomorphism $φ: Hk(q) → Tk(n) given by φ(Ti) = Ei-q-1 and φ(q+q-1) = n,$ with $kerφ = ⟨ TiTi+1Ti + TiTi+1 + Ti+1Ti + Ti + Ti+1 + 1 ⟩.$ Composing with the surjective homomorphism $H˜k(q) → Hk(q) Xεi ↦ Ti-1 ⋯ T2T11T2⋯Ti-1 Ti ↦ Ti$

### Murphy elements

Let us write $Ti = Ei-q-1, so that Xε1 = 1, and Xεi = Ti-1 Xεi-1 Ti-1$ in the Temperley-Lieb algebra. Then define ${m}_{1},...,{m}_{k}$ by Solving for ${X}^{{\epsilon }_{i}}$ in terms of the ${m}_{i}$ gives $Xεi = (q-q-1) ( q-(i-2)mi + q-(i-2+1)mi-1 + ⋯ + q-(2i-4)m2 ) + q-2(i-1),$ from which one obtains $q(k-2) ( Xε1 + Xε2 +⋯+ Xεk ) - q[k] = (q-q-1) ( mk+[2]mk-1 +⋯+ [k-1]m2 ).$ Using the definition of ${X}^{{\epsilon }_{i}}$ and substituting for ${X}^{{\epsilon }_{i-1}}$ in terms of the ${m}_{i}$ gives $(q-q-1)mi = qi-2 Xεi - qi-4 Xεi-1 = qi-2 ( Ei-1-q-1 ) Xεi-1 ( Ei-1-q-1 ) - qi-4 Xεi-1 = qi-2 Ei-1 Xεi-1 Ei-1 - qi-3 ( Ei-1 Xεi-1 + Xεi-1 Ei-1 ) = qi-2 Ei-1 ( ( q-q-1 ) ( q-(i-3)mi + q-(i-3+1)mi-1 + ⋯ + q-(2i-6)m2 ) + q-2(i-2) ) Ei-1 - qi-3 Ei-1 ( ( q-q-1 ) ( q-(i-3)mi + q-(i-3+1)mi-1 + ⋯ + q-(2i-6)m2 ) + q-2(i-2) ) - qi-3 ( ( q-q-1 ) ( q-(i-3)mi + q-(i-3+1)mi-1 + ⋯ + q-(2i-6)m2 ) + q-2(i-2) ) Ei-1 = qi-2 (q-q-1) q-(i-3) Ei-1 mi-1 Ei-1 - qi-3 (q-q-1) q-(i-3) ( Ei-1 mi-1 + mi-1 Ei-1 ) + qi-2 (q+q-1) Ei-1 ( ( q-q-1 ) ( q-(i-3)mi + q-(i-3+1)mi-1 + ⋯ + q-(2i-6)m2 ) + q-2(i-2) ) -2qi-3 Ei-1 ( ( q-q-1 ) ( q-(i-3)mi + q-(i-3+1)mi-1 + ⋯ + q-(2i-6)m2 ) + q-2(i-2) ) = qi-2 (q-q-1) q-(i-3) Ei-1 mi-1 Ei-1 - qi-3 (q-q-1) q-(i-3) ( Ei-1 mi-1 + mi-1 Ei-1 ) + qi-2 (q-q-1) Ei-1 ( ( q-q-1 ) ( q-(i-3)mi + q-(i-3+1)mi-1 + ⋯ + q-(2i-6)m2 ) + q-2(i-2) )$ since ${E}_{i-1}$ commutes with ${m}_{2},{m}_{3},...,{m}_{i-1}.$ Thus $mi = q-(i-2) Ei-1 + qEi-1 mi-1 Ei-1 - ( Ei-1 mi-1 + mi-1 Ei-1 ) + (q-q-1) ( mi-2 + q-1 mi-3 + q-2 mi-4 + ⋯ + q-(i-4)m2 ) Ei-1.$ It seems to me that this formula provides the easiest way to compute ${m}_{i}$ in terms of the $E$s. I would not be too worried about the coefficients of ${E}_{1}{E}_{4}$ and ${E}_{2}{E}_{4}$ in ${m}_{4}$ looking strange. One expects diagrams that are equal to their own flip to act a bit differently in ${m}_{k}.$ Note also that $[3]-1 = [4] [2] and [3]+1 = [2]2,$ so these are pretty nice $q-$versions of 2. Let's have a look at ${m}_{6}$ and see if we can get an induction going. It might help to categorize the terms according to what their flip is to see where the next level is coming from.

For $n$ such that $ℂ{T}_{k}\left(n\right)$ is semisimple, the simple ${T}_{k}\left(n\right)$ are indexed by partitions in the set $T^k = {λ⊢k | λ has at most two columns }.$ The irreducible $ℂ{T}_{k}\left(n\right)$ modules have seminormal basis ${vT | T is a standard tableau of shape λ}$ and $XεivT = q2c(T(i)) vT.$ Since $c\left(T\left(i\right)\right)=c\left(T\left(i-1\right)\right)-1$ if the boxes $T\left(i\right)$ and $T\left(i-1\right)$ are in the same column and $c\left(T\left(i\right)\right)+c\left(T\left(i-1\right)\right)=3-i$ if the boxes $T\left(i\right)$ and $T\left(i-1\right)$ are in different columns it follows that $mivT = qi-2q2c(T(i)) - qi-4 q2c(T(i-1)) q-q-1 = cT(i) vT,$ where $cT(i) = { 0, if T(i) and T(i-1) are in the same column, [ i-2+2 c(T(i)) ], if T(i) and T(i-1) are in different columns. }$

Now we want to define pseudomatrix units in $ℂ{T}_{k}\left(n\right)$ according to the left and right eigenspaces of the ${m}_{i}.$ Let $pST ∈ LS∩RT,$ normalized so that the coefficients are in $ℤ\left[n\right]$ with greatest common divisor 1. Then $pST pUV = γT δUV pSV, pST = ∑S+,T+ cS+T+ pS+T+, pST ek pUV = βT- δT-U- pp+T+, ek+1 pST ek+1 = εS+T+ δS(k)T(k) pST ek+1.$

### Examples

Let's start with generic $n.$ Here $eST = [a] [b] eS-U- Ek-1 eU-T-.$ Then $Ek = ∑ [b] [a] eST and mk = ∑μk(S)eSS,$ where the first sum is over all pairs $\left(S,T\right)$ such that $S=T$ or $S$ and $T$ differ at the $k-1$st level.

In $ℂ{T}_{2}\left(n\right)$ let $p12,12 p 12 , 12 = [2]e12,12 [2]e 12 , 12$ In $ℂ{T}_{3}\left(n\right)$ let $p 1 2 3 , 1 2 3 p 1 2 3 , 1 3 2 p 1 3 2 , 1 2 3 p 1 3 2 , 1 3 2 p 1 2 3 , 1 2 3 = [2]e 1 2 3 , 1 2 3 [3][2] e 1 2 3 , 1 3 2 [2]e 1 3 2 , 1 2 3 [3][2] e 1 3 2 , 1 3 2 [3] e 1 2 3 , 1 2 3$ In $ℂ{T}_{4}\left(n\right)$ let $p 1 2 3 4 , 1 2 3 4 p 1 2 3 4 , 1 3 2 4 p 1 3 2 4 , 1 2 3 4 p 1 3 2 4 , 1 3 2 4 p 1 2 3 3 , 1 2 3 3 p 1 2 3 3 , 1 3 2 2 p 1 2 3 3 , 1 4 2 2 p 1 3 2 2 , 1 2 3 3 p 1 3 2 2 , 1 3 2 2 p 1 3 2 2 , 1 4 2 2 p 1 4 2 2 , 1 2 3 3 p 1 4 2 2 , 1 3 2 2 p 1 4 2 2 , 1 4 2 2 p 1 2 3 4 , 1 2 3 4 = [2]2 e 1 2 3 4 , 1 2 3 4 [2]2 e 1 2 3 4 , 1 3 2 4 [2]2 e 1 3 2 4 , 1 2 3 4 [2]2 e 1 3 2 4 , 1 3 2 4 [3][2]2 e 1 2 3 3 , 1 2 3 3 [3][2]2 e 1 2 3 3 , 1 3 2 2 [3][2]2 e 1 2 3 3 , 1 4 2 2 [3][2]2 e 1 3 2 2 , 1 2 3 3 [3][2]2 e 1 3 2 2 , 1 3 2 2 [3][2]2 e 1 3 2 2 , 1 4 2 2 [3][2]2 e 1 4 2 2 , 1 2 3 3 [3][2]2 e 1 4 2 2 , 1 3 2 2 [3][2]2 e 1 4 2 2 , 1 4 2 2 [4][3][2] e 1 2 3 4 , 1 2 3 4$

The special value $n=±\sqrt{2},$ i.e. when $\left[4\right]=0.$
Then $p 1 4 2 2 , 1 4 3 2 = p 1 2 3 3 , 1 2 3 3 and we let p 1 4 2 2 , 1 4 2 2 (2) = 1- e 1 2 3 , 1 2 3 .$ In this basis $Rad(ℂT4) = span 0 0 0 0 0 0 1 0 0 1 1 1 1 0 and Rad2(ℂT4) = span 0 0 0 0 0 0 0 0 0 0 0 0 1 0$ $ℂT1 = {(a)} = { a a } = { a 0 0 a a }$ and $ℂT2 = { a1 a2 } = { a1 0 0 a2 a2 }$ and $ℂT3 = { a11 a12 a21 a22 a3 } = { a11 a12 a21 a22 a11 a12 0 a21 a22 0 0 0 a3 a3 }.$

The special value $n=±1,$ i.e. when $\left[3\right]=0.$
Then $p 1 3 2 , 1 3 2 = p 1 2 3 , 1 2 3 and we let p 1 3 2 , 1 3 2 (2) = 1- e 1 2 3 , 1 2 3 .$ In this basis $Rad(ℂT3) = span 0 1 1 1 0 and Rad2(ℂT3) = span 0 0 1 0 0 .$ Then $E1 = 1 0 0 0 0 , E2 = 1 1 1 1 0 , 1 = 1 0 0 0 1 ,$ $m2 = 1 0 0 0 0 , m3 = -1 0 0 1 0 .$ $ℂT1 = { a a } = { a 0 0 0 a }$ and $ℂT2 = { a1 a2 } = { a1 0 0 0 a2 }.$

The special value $n=0,$ i.e. when $\left[2\right]=0.$
Then $p12,12 = p 12 , 12 and we let p 12,12 (2) = 1.$ In the basis $p12,12 p12,12(2)$ $e1 = 1 0 , m2 = 1 0 , and Rad(ℂT2) = span 1 0 .$ With respect to this basis there is a new matrix $ℰ = e2p12,122e2 e2p12,12 p12,12(2)e2 e2p12,12(2)p12,12e2 e2 ( p12,12(2) )2e2 = n 1 1 n = 0 1 1 0 ,$ which is not diagonal. In $ℂ{T}_{3}$ the basis elements %%%%%%%%%%%%%%%% $p 1 2 3 , 1 2 3 (2) p 1 2 3 , 1 3 2 p 1 3 2 , 1 2 3 (2) p 1 3 2 , 1 3 2 (2) p 1 2 3 , 1 2 3 = p12,12 e2 p12,12(2) p12,12(2) e2 p12,12 (2) p12,12 e2 p12,12 p12,12 e2 p12,12(2) 1- p 1 2 3 , 1 2 3 (2) - p 1 3 2 , 1 3 2 (2)$ form a set of matrix units. In this basis $E1 = 0 1 0 0 0 , E2 = 0 0 1 0 0 , 1 = 1 0 0 1 1 ,$ $m2 = 0 1 0 0 0 , m3 = -1 0 0 -1 0 ,$ $ℂT1 = {(a)} = { 0 a } = { a 0 0 a a }$ and $ℂT2 = { a2 a1 } = { a1 a2 0 a1 a1 }.$

## References

[GW1] F. Goodman and H. Wenzl, The Temperley-Lieb algebra at roots of unity, Pacific J. Math. 161 (1993), no. 2, 307-334.

[GL1] J. Graham and G. Lehrer, Diagram algebras, Hecke algebras and decomposition numbers at roots of unity, Ann. Sci. École Norm. Sup. (4) 36 (2003), no. 4, 479-524.

[GL2] J. Graham and G. Lehrer, The two-step nilpotent representations of the extended affine Hecke algebra of type A, Compositio Math. 133 (2002), no. 2, 173-197.