## A Ribbon Hopf Algebra Approach to the Irreducible Representations of Centralizer Algebras: The Brauer, Birman-Wenzl, and Type A Iwahori-Hecke Algebras

Last update: 7 March 2014

## Path Algebras and Tensor Power Centralizer Algebras

Bratteli Diagrams

A Bratteli diagram $A$ is a graph with vertices from a collection of sets ${\stackrel{ˆ}{A}}_{m},$ $m\ge 0,$ and edges that connect vertices in ${\stackrel{ˆ}{A}}_{m}$ to vertices in ${\stackrel{ˆ}{A}}_{m+1}\text{.}$ We assume that the set ${\stackrel{ˆ}{A}}_{0}$ contains a unique vertex denoted $\varnothing \text{.}$ It is possible that there are multiple edges connecting any two vertices. We shall call the vertices shapes. The set ${\stackrel{ˆ}{A}}_{m}$ is the set of shapes on level $m\text{.}$ If $\lambda \in {\stackrel{ˆ}{A}}_{m}$ is connected by an edge to a shape $\mu \in {\stackrel{ˆ}{A}}_{m+1}$ we write $\lambda \le \mu \text{.}$

A multiplicity free Bratteli diagram is a Bratteli diagram such that there is at most one edge connecting any two vertices. Alternatively we could define a multiplicity free Bratteli diagram to be a ranked poset $A$ which is ranked by the nonnegative integers and such that there is a unique vertex on level $0$ called $\varnothing \text{.}$ Identifying the poset $A$ with its Hasse diagram we see that these two definitions are the same since the poset condition implies that the resulting Bratteli diagram is multiplicity free. In order to make sure that we do not make careless statements in this paper. $Assume throughout this paper that all Bratteli diagrams are multiplicity free.$ We make this assumption to simplify our proofs and our notation. See [GHJ1989] for the more general setting.

The Bratteli diagrams which we will be most interested in, see Figures 1 and 2, are multiplicity free and arise naturally in the representation theory of centralizer algebras. Other examples of Bratteli diagrams arise from differential posets [Sta1988] and towers of ${C}^{*}$ algebras [GHJ1989]. The Bratteli diagrams in Figures 1 and 2 are described further in Sections 4 and 5 respectively.

Paths and Tableaux

Let $A$ be a multiplicity free Bratteli diagram and let $\lambda \in {\stackrel{ˆ}{A}}_{m}$ and $\mu \in {\stackrel{ˆ}{A}}_{n}$ where $m A path from $\lambda$ to $\mu$ is a sequence of shapes ${\lambda }^{\left(i\right)},$ $m\le i\le n,$ $P= ( λ(m), λ(m+1),…, λ(n) )$ such that $\lambda ={\lambda }^{\left(m\right)}\le {\lambda }^{\left(m+1\right)}\le \cdots \le {\lambda }^{\left(n\right)}=\mu$ and ${\lambda }^{\left(i\right)}\in {\stackrel{ˆ}{A}}_{i}\text{.}$ In the poset sense the path $P$ is a saturated chain from $\lambda$ to $\mu \text{.}$ (If we are working in the nonmultiplicity free setting we must distinguish paths which "travel" from ${\lambda }^{\left(i\right)}$ to ${\lambda }^{\left(i+1\right)}$ along different edges.) A tableau $T$ of shape $\lambda$ is a path from $\varnothing$ to $\lambda$ $T= ( λ(0), λ(1),…, λ(m) )$ such that $\varnothing ={\lambda }^{\left(0\right)}\le {\lambda }^{\left(1\right)}\le \cdots \le {\lambda }^{\left(m\right)}=\lambda$ and ${\lambda }^{\left(i\right)}\in {\stackrel{ˆ}{A}}_{i}$ for each $1\le i\le m\text{.}$ We write $\text{shp}\left(T\right)=\lambda$ if $T$ is a tableau of shape $\lambda \text{.}$ We say that the length of $T$ is $m$ if $\text{shp}\left(T\right)\in {\stackrel{ˆ}{A}}_{m}\text{.}$

Let us make the following (hopefully suggestive) notations. ${𝒯}^{\lambda }$ is the set of tableaux of shape $\lambda ,$
${𝒯}^{m}$ is the set of tableaux of length $m,$
${𝒯}_{\lambda }^{\mu }$ is the set of paths from $\lambda$ to $\mu ,$
${𝒯}_{\lambda }^{m}$ is the set of paths from $\lambda$ to any shape on level $m,$
${𝒯}_{T}^{m}$ is the set of paths from $\text{shp}\left(T\right)$ to any shape on level $m\text{.}$
Similarly, we define ${\Omega }^{\lambda }$ is the set of pairs $\left(S,T\right)$ of paths $S,T\in {𝒯}^{\lambda },$
${\Omega }^{m}$ is the set of pairs $\left(S,T\right)$ of paths $S,T\in {𝒯}^{m}$ such that $\text{shp}\left(S\right)=\text{shp}\left(T\right),$
${\Omega }_{\lambda }^{\mu }$ is the set of pairs $\left(S,T\right)$ of paths $S,T\in {𝒯}_{\lambda }^{\mu },$
${\Omega }_{\lambda }^{m}$ is the set of pairs $\left(S,T\right)$ of paths $S,T\in {𝒯}_{\lambda }^{m}$ such that $\text{shp}\left(S\right)=\text{shp}\left(T\right)\text{.}$

Path Algebras

For each $m$ define an algebra ${A}_{m}$ over a field $k$ with basis ${E}_{ST},$ $\left(S,T\right)\in {\Omega }^{m}$ and multiplication given by $ESTEPQ= δTPESQ. (1.1)$ Note that ${A}_{0}\simeq k\text{.}$ Every element $a\in {A}_{m}$ can be written in the form $a=∑(S,T)∈Ωm aSTEST,$ for some constants ${a}_{ST}\in k\text{.}$ In this way we can view each element $a\in {A}_{m}$ as weighted sum of pairs of paths, where the weight of a pair of paths $\left(S,T\right)\in {\Omega }^{m}$ is the constant ${a}_{ST}\text{.}$ We shall refer to the collection of algebras ${A}_{m}$ as the tower of path algebras corresponding to the Bratteli diagram $A\text{.}$

Each of the algebras ${A}_{m}$ is isomorphic to a direct sum of matrix algebras $Am≃⨁λ∈Aˆm Mdλ(k),$ where ${M}_{d}\left(k\right)$ denotes the algebra of $d×d$ matrices with entries from $k$ and ${d}_{\lambda }=\text{Card}\left({𝒯}^{\lambda }\right)\text{.}$ Thus, the irreducible representations of ${A}_{m}$ are indexed by the elements of ${\stackrel{ˆ}{A}}_{m}\text{.}$ Furthermore, the dimensions of these irreducible representations are equal to $\text{Card}\left({𝒯}^{\lambda }\right),$ and thus, the set of tableaux ${𝒯}^{\lambda }$ is a natural index set for a basis of the irreducible ${A}_{m}\text{-module}$ indexed by $\lambda \in {\stackrel{ˆ}{A}}_{m}\text{.}$

The Inclusions ${A}_{m}\subseteq {A}_{n},$ $m\le n$

Given a path $T=\left(\lambda ,\dots ,\mu \right)$ from $\lambda$ to $\mu$ and a path $S=\left(\mu ,\dots ,\nu \right)$ from $\mu$ to $\nu$ define $T*S=(λ,…,μ,…,ν) (1.2)$ to be the concatenation of the two paths (the shape $\mu$ is not repeated since that would not produce a path).

Let $0\le m Define an inclusion of ${A}_{m}\subseteq {A}_{n}$ as follows: For each $\left(P,Q\right)\in {\Omega }^{m}$ view ${E}_{PQ}$ as an element of ${A}_{n}$ by the formula $EPQ=∑T∈𝒯λn EP*T,Q*T,where λ=shp(P)=shp(Q). (1.3)$ In particular we have an inclusion of ${A}_{m-1}$ into ${A}_{m}$ for every $m>0\text{.}$ Let $\lambda \in {\stackrel{ˆ}{A}}_{m}$ and let ${V}^{\lambda }$ be the irreducible representation of ${A}_{m}$ corresponding to $\lambda \text{.}$ Then the restriction of ${V}^{\lambda }$ to ${A}_{m-1}$ decomposes as $Vλ↓Am-1Am ≃⨁μ∈λ-Vμ,$ where ${\lambda }^{-}=\left\{\mu \in {\stackrel{ˆ}{A}}_{m-1} | \mu \le \lambda \right\}\text{.}$ The multiplicity free condition on the Bratteli diagram guarantees that this decomposition is multiplicity free.

The Centralizer of ${A}_{m}$ Contained in ${A}_{n},$ $0\le m

Define $𝒵(Am⊆An)= { a∈An | ab=ba for all b∈Am } .$ Let us extend the notation in (1.3) and define $EST=∑P∈𝒯λ EP*S,P*T,$ for each pair $\left(S,T\right)\in {\Omega }_{\lambda }^{\mu },$ $\lambda \in {\stackrel{ˆ}{A}}_{m},$ $\mu \in {\stackrel{ˆ}{A}}_{n},$ the following result appears in [GHJ1989, Proposition 2.3.12].

The elements ${E}_{ST},$ $\left(S,T\right)\in {\Omega }_{\lambda }^{\mu },$ $\lambda \in {\stackrel{ˆ}{A}}_{m},$ $\mu \in {\stackrel{ˆ}{A}}_{n},$ are a basis of $𝒵\left({A}_{m}\subseteq {A}_{n}\right)\text{.}$

 Proof. First let us show that the elements ${E}_{ST}\in 𝒵\left({A}_{m}\subseteq {A}_{n}\right)\text{.}$ Let $\gamma \in {\stackrel{ˆ}{A}}_{m}$ and let $Q,R\in {𝒯}^{\gamma }\text{.}$ Then $ESTEQR = ( ∑P∈𝒯λ EP*S,P*T ) ( ∑U∈𝒯γn EQ*U,R*U ) = EQ*S,Q*T ( ∑U∈𝒯γn EQ*U,R*U ) = EQ*S,Q*T EQ*T,R*T= EQ*S,R*T.$ Similarly one shows that ${E}_{QR}{E}_{ST}={E}_{Q*S,R*T},$ giving that ${E}_{ST}\in 𝒵\left({A}_{m}\subseteq {A}_{n}\right)\text{.}$ Now we show that if $a\in 𝒵\left({A}_{m}\subseteq {A}_{n}\right)$ then $a$ is a linear combination of ${E}_{ST}\text{.}$ Suppose $a=∑(M,N)∈Ωn aMNEMN∈𝒵 (Am⊆An).$ Let $\lambda \in {\stackrel{ˆ}{A}}_{m}$ and let $P\in {𝒯}^{\lambda }\text{.}$ Then $aEPP = ( ∑(M,N)∈Ωn aMNEMN ) ( ∑T∈𝒯λn EP*T,P*T ) = ∑(M,P*T)∈Ωn aM,P*T EM,P*T EPPa = ( ∑S∈𝒯λn EP*S,P*S ) ( ∑(M,N)∈Ωn aMNEMN ) = ∑(P*S,N)∈Ωn aP*S,N EP*S,N$ This implies that ${a}_{M,P*T}=0$ unless $M=P*S$ for some $S\in {𝒯}_{\lambda }^{n}$ and ${a}_{P*S,N}=0$ unless $N=P*T,$ for some $T\in {𝒯}_{\lambda }^{n}\text{.}$ Thus, $a$ must be of the form $a=∑P∈𝒯m(S,T)∈Ωλn aP*S,P*T EP*S,P*T.$ If $\lambda \in {\stackrel{ˆ}{A}}_{m}$ and $\left(P,Q\right)\in {\Omega }^{\lambda }$ then $EPQa = ( ∑S∈𝒯λn EP*S,Q,S ) ( ∑R∈𝒯m(S,T)∈Ωλn aR*S,R*T ER*S,R*T ER*S,R*T ) = ∑(S,T)∈Ωλn aQ*S,Q*T EP*S,Q*T, aEPQ = ( ∑R∈𝒯m(S,T)∈Ωλn aR*S,R*T ER*S,R*T ) ( ∑T∈𝒯λn EP*T,Q*T ) = ∑(S,T)∈Ωλn aP*S,P*T EP*S,Q*T.$ This implies that ${a}_{Q*S,Q*T}={a}_{P*S,P*T}$ for all $\left(P,Q\right)\in {\Omega }^{\lambda }\text{.}$ Let us denote this coefficient by ${a}_{ST}\text{.}$ Then $a=∑λ∈Aˆm(S,T)∈Ωλm aST∑P∈𝒯λ EP*S,P*T= ∑λ∈Aˆm(S,T)∈Ωλm aSTEST.$ Thus, if $a\in 𝒵\left({A}_{m}\subseteq {A}_{n}\right)$ then $a$ is a linear combination of ${E}_{ST}\text{.}$ The elements ${E}_{ST},$ $\left(S,T\right)\in {\Omega }_{m}^{n}$ are independent since the elements ${E}_{MN},$ $\left(M,N\right)\in {\Omega }^{n}$ are. $\square$

Let ${A}_{m},$ $m\ge 0,$ be the tower of path algebras corresponding to a multiplicity free Bratteli diagram A and suppose that ${g}_{i}\in {A}_{i+1},$ $i\ge 1,$ are elements such that

 (1) For each $m,$ the elements ${g}_{1},{g}_{2},\dots ,{g}_{m-1}$ generate ${A}_{m},$ (2) ${g}_{i}{g}_{j}={g}_{j}{g}_{i}$ for all $i,j$ such that $|i-j|>1\text{.}$
Then $gm-1=∑(P,Q)∈Ωm-2m (gm-1)PQ EPQ$ for some constants ${\left({g}_{m-1}\right)}_{PQ}\in k\text{.}$

 Proof. It follows from the relations on the ${g}_{i}$ that ${g}_{m-1}$ commutes with ${A}_{m-2}\text{.}$ The result then follows from Proposition (1.4). $\square$

Let ${A}_{m},$ $m\ge 0,$ be the tower of path algebras corresponding to a multiplicity free Bratteli diagram A and suppose that ${g}_{i}\in {A}_{i+1},$ $i\ge 1,$ are elements such that

 (1) For each $m,$ the elements ${g}_{1},{g}_{2},\dots ,{g}_{m-1}$ generate ${A}_{m},$ (2) ${g}_{i}{g}_{j}={g}_{j}{g}_{i}$ for all $i,j$ such that $|i-j|>1\text{.}$ (3) ${g}_{i}{g}_{i+1}{g}_{i}={g}_{i+1}{g}_{i}{g}_{i+1}$ for all $i\ge 1\text{.}$
Define ${D}_{m}={g}_{m-1}{g}_{m-2}\cdots {g}_{2}{g}_{1}{g}_{1}{g}_{2}\cdots {g}_{m-3}{g}_{m-2}{g}_{m-1}\in {A}_{m}\text{.}$ Then $Dm=∑S∈𝒯m-1m DSSESS,$ for some constants ${D}_{SS}\in k\text{.}$

 Proof. Using the braid relation (3) for the elements ${g}_{i}$ we have $gm-2Dm = gm-1gm-2 gm-1gm-3 ⋯g1g1⋯ gm-1 = gm-1gm-2 gm-3⋯g1g1 ⋯gm-1gm-2 gm-1 = Dmgm-2.$ It follows that ${D}_{m}$ commutes with ${g}_{m-2}\text{.}$ By induction we have that $gjDm = gm-1⋯ gj+2gj Dj+2 gj+2 gj+3⋯ gm-1 = gm-1⋯ gj+2 Dj+2gj gj+2 gj+3⋯ gm-1 = Dmgj,$ for all $1\le j Thus, ${D}_{m}$ commutes with ${A}_{m-1}\text{.}$ The result now follows from Proposition (1.4). $\square$

Remark. All of the above results hold even if the Bratteli diagram is not multiplicity free since the main result Proposition (1.4) holds in that case. We have stated these results only for the multiplicity free case in order to simplify our notation. See [GHJ1989] for the more general setting.

Centralizers of Tensor Power Representations

Let $k$ be a field. We shall assume that $k$ is characteristic zero and algebraically closed. Let $𝔘$ be a Hopf algebra over $k$ such that all finite dimensional representations of $𝔘$ are completely reducible. Let $V$ be a finite dimensional representation of $𝔘$ and define $𝒵m=End𝔘 (V⊗m). (1.7)$ Let $\stackrel{ˆ}{𝔘}$ be an index set for the finite dimensional irreducible representations of $𝔘\text{.}$ Let ${\stackrel{ˆ}{𝒵}}_{m}$ be an index set for the finite dimensional representations of ${𝒵}_{m}\text{.}$ It is natural to view ${\stackrel{ˆ}{𝒵}}_{m}$ as a subset of $\stackrel{ˆ}{𝔘}$ since, by Schur-Weyl duality, the $\left({𝒵}_{m}\otimes 𝔘\right)\text{-module}$ ${V}^{\otimes m}$ has a decomposition $V⊗m≅ ⨁λ∈𝒵ˆm 𝒵λ⊗Λλ,$ where ${𝒵}^{\lambda }$ is the irreducible ${𝒵}_{m}\text{-module}$ indexed by $\lambda$ and ${\Lambda }_{\lambda }$ is the irreducible $𝔘\text{-module}$ indexed by $\lambda \text{.}$

For $0 there is a natural inclusion ${𝒵}_{m}\subset {𝒵}_{n}$ given by $𝒵m ↪ 𝒵n a ↦ a⊗id⊗(n-m)$ where $a\otimes {\text{id}}^{\otimes \left(n-m\right)}$ acts as $a$ on the first $m$ factors of ${V}^{\otimes n}$ and as the identity on the last $m-n$ tensor factors. By convention we shall set ${𝒵}_{0}=k\text{.}$ If $V$ is an irreducible $𝔘\text{-module}$ then, by Schur's lemma, ${𝒵}_{1}\cong k\text{.}$

The Bratteli Diagram for Tensor Powers of $V$

Assume that $V$ is an irreducible $𝔘\text{-module.}$ Let $\lambda \in {\stackrel{ˆ}{𝒵}}_{m}$ for some $m\text{.}$ Then there is a branching rule for tensoring by $V$ which describes the decomposition $Λλ⊗V= ⨁μ∈𝒵ˆm+1 cλVμΛμ, (1.8)$ as $𝔘\text{-modules.}$ The multiplicities ${c}_{\lambda V}^{\nu }$ are nonnegative integers. This decomposition is multiplicity free if all the multiplicities ${c}_{\lambda V}^{\nu }\le 1\text{.}$ Let $\nu \in {\stackrel{ˆ}{𝒵}}_{m+1}\text{.}$ Then the branching rule for inclusion ${𝒵}_{m}\subseteq {𝒵}_{m+1}$ describes the decomposition $𝒵ν=⨁λ∈𝒵ˆm cλVν𝒵λ, (1.9)$ as ${𝒵}_{m}\text{-modules.}$ There is a standard reciprocity result for branching rules ([Bou1981] Chpt. VIII §5 Ex. 17, see also [Ram1991-4] Theorem 5.9 for a simple proof), that states that the constants ${c}_{\lambda V}^{\nu }$ appearing in (1.8) and (1.9) are the same.

We define a Bratteli diagram for tensor powers of $V,$ or equivalently, a Bratteli diagram for the tower of algebras ${𝒵}_{m},$ as follows. Let the elements of the set ${\stackrel{ˆ}{𝒵}}_{m}$ be the vertices on level $m\text{.}$ A vertex $\lambda \in {\stackrel{ˆ}{𝒵}}_{m}$ is connected to a vertex $\mu \in {\stackrel{ˆ}{𝒵}}_{m+1}$ by ${c}_{\lambda V}^{\mu }$ edges. This Bratteli diagram is multiplicity free if the corresponding branching rule for tensoring by $V$ is multiplicity free.

Identification of the Centralizer Algebras ${𝒵}_{m}$ with Path Algebras

By working inductively, we can view the algebras ${𝒵}_{m}$ as path algebras for the Bratteli diagram for tensor powers of $V\text{.}$ Let us denote this Bratteli diagram by $A$ and denote the corresponding path algebras by ${A}_{m}\text{.}$ Clearly ${𝒵}_{0}\cong k$ can be identified with the corresponding path algebra ${A}_{0}\text{.}$ For each $\lambda \in \stackrel{ˆ}{𝔘}$ let ${\Lambda }_{\lambda }$ denote the irreducible $𝔘$ module corresponding to $\lambda \text{.}$ Suppose that there is an identification of ${𝒵}_{m}$ with the path algebra ${A}_{m}$ so that $V⊗m=⨁λ∈𝒵ˆm ( ⨁T∈𝒯λ ETTV⊗m ) ,$ is a decomposition of ${V}^{\otimes m}$ so that the $𝔘\text{-submodule}$ ${E}_{TT}{V}^{\otimes m}\cong {\Lambda }_{\lambda }\text{.}$ The element ${E}_{TT}$ is a $𝔘\text{-invariant}$ projection onto the irreducible $𝔘\text{-module}$ ${E}_{TT}{V}^{\otimes m}\text{.}$

Given a tableau $T=\left({\tau }^{\left(0\right)},\dots ,{\tau }^{\left(m-1\right)},\lambda \right)\in {𝒯}^{\lambda }$ and a shape $\nu \in {𝒵}_{m+1}$ such that $\nu \ge \lambda$ let $T*\nu$ be the path given by $T*\nu =\left({\tau }^{\left(0\right)}\right),\dots ,{\tau }^{\left(m-1\right)},\lambda ,\nu \text{.}$ Since the branching rule for tensoring by $V$ is multiplicity free, there is a unique decomposition $(ETTV⊗m) ⊗V=⨁ν∈𝒵ˆm+1ν≥λ VT*ν, (1.10)$ into nonisomorphic irreducible $𝔘\text{-modules}$ ${V}_{T*\nu }\cong {\Lambda }_{\nu }\text{.}$ Define ${E}_{T*\nu ,T*\nu }\in {𝒵}_{m+1}$ to be the unique $𝔘\text{-invariant}$ projection onto the irreducible ${V}_{T*\nu }$ in the decomposition (1.10). In this way we can define elements ${E}_{SS}$ for every $S\in {𝒯}^{m+1}$ and we have that $V⊗(m+1)= ⨁ν∈𝒵ˆm+1 ( ⨁S∈𝒯ν ESSV⊗m ) ,$ is a decomposition of ${V}^{\otimes \left(m+1\right)}$ into irreducible $𝔘\text{-modules}$ ${E}_{SS}{V}^{\otimes \left(m+1\right)}\cong {\Lambda }_{\nu },$ $S\in {𝒯}^{\nu }\text{.}$ This makes an identification of each basis element ${E}_{SS},$ $S\in {𝒯}^{m+1},$ of the path algebra ${A}_{m+1}$ with a transformation in ${𝒵}_{m+1}\text{.}$ Now, for each pair of paths $\left(P,Q\right)\in {\Omega }^{m+1}$ choose nonzero transformations $EPQ∈EPP 𝒵m+1EQQ andEQP∈EQQ 𝒵m+1EPP$ and normalize them so that $EPQEQP= EPP, (1.11)$ as transformations in ${𝒵}_{m+1}\text{.}$ In this way, one can identify the path algebra ${A}_{m+1}$ with the algebra ${𝒵}_{m+1}\text{.}$ This identification is not canonical, there is the following freedom in the choice of the normalization of the transformations ${E}_{PQ}$ and ${E}_{QP}\text{:}$ For any nonzero constant $\alpha \in k,$ one may $replace EPQ and EQP by α EPQ and (1/α) EPQ respectively, (1.12)$ to get another solution.

Suppose that an identification of the centralizer algebras ${𝒵}_{m}$ with the path algebras is given. This identification determines a choice of the irreducible representations of ${𝒵}_{m}$ in the following way. If $a\in {𝒵}_{m},$ and $a=∑λ∈𝒵ˆm ∑(S,T)∈Ωλ (a)STEST,$ then the maps $πλ: 𝒵m ⟶ Mdλ(k) a ⟼ ((a)ST)(S,T)∈Ωλ$ for $\lambda \in {\stackrel{ˆ}{𝒵}}_{m},$ determine a complete set of nonisomorphic irreducible representations of ${𝒵}_{m}\text{.}$ In this paper we shall find path algebra formulas for the generators of tensor power centralizer algebras, ${𝒵}_{m},$ and thus, in essence, we are finding the irreducible representations.

## Notes and References

This is a typed version of the paper A Ribbon Hopf Algebra Approach to the Irreducible Representations of Centralizer Algebras: The Brauer, Birman-Wenzl, and Type A Iwahori-Hecke Algebras by Robert ${\text{Leduc}}^{*}$ and Arun ${\text{Ram}}^{†}\text{.}$

The paper was received June 24, 1994; accepted September 12, 1994.

${}^{*}$Supported in part by National Science Foundation Grant DMS-9300523 to the University of Wisconsin.
${}^{†}$Supported in part by National Science Foundation Postdoctoral Fellowship DMS-9107863.