considered here are ﬁnite dimensional associative algebras over k, with a unit element 1. A partition of unity in A is a complete set of pairwise primitive orthogonal idempotents that sum to 1. We study the following two important (related) problems: Find a partition of unity in A. Find the (unique) partition of unity in its center Z(A). These are fundamental problems for any A. We solve them only in a special case, in which the given algebra A = An ﬁts into a multiplicity-free family (MFF). The collection {An | n ≥ 0} is an MFF if each Ak is split semisimple over k, and: 1 A0 ∼ = k. 2 There is an embedding Ak ⊆ Ak+1 sending 1 to 1, for all k ≥ 0. 3 For any given irrep V of Ak, the multiplicities of its irreducible summands on restriction to Ak−1 are all ≤ 1.
| n ≥ 0} of symmetric group algebras over a ﬁeld k of characteristic 0. Other examples of MFFs include: The Iwahori–Hecke algebras {Hq(Sn) | n ≥ 0}. Certain families of other Weyl groups and their Iwahori–Hecke algebras (see [Ram 97]). The partition algebras. The Brauer algebras and Birman–Murakami–Wenzl (BMW) algebras. Etc, . . . Note that in the case of diagram algebras there are restrictions on the parameters as well as the underlying ﬁeld in order to ensure semisimplicity.
an MFF. Write Irr(n) for a complete set of isomorphism classes of irreducible An-modules. Pick a representative V λ in each class λ ∈ Irr(n). The branching graph (Bratelli diagram) B is the directed graph with vertices and edges as follows: the vertices are the isomorphism classes n≥0 Irr(n); there is an edge µ → λ from the vertex µ to the vertex λ if and only if V µ is isomorphic to a direct summand of the restriction of V λ. Given λ ∈ Irr(n), let Tab(λ) denote the set of paths in the branching graph starting from the unique element ∅ ∈ Irr(0) and terminating at λ. An element of Tab(λ) has the form T : ∅ = λ0 → λ1 → λ2 → · · · → λn−1 → λn = λ. Set Tab(n) = λ∈Irr(n) Tab(λ). We say that a path T ∈ Tab(n) has length n. We sometimes write T → λ to indicate that T ∈ Tab(λ).
By the multiplicity-free branching assumption and Schur’s Lemma, the decomposition resAn−1 V = [W ]→[V ] W is canonical. Decomposing each W on the right hand side upon restriction to An−2 and continuing inductively all the way down to A0 ∼ = k, we obtain a canonical decomposition resA0 V = T VT into irreducible A0-modules, which are the 1-dimensional subspaces VT, where T runs over the set of T ∈ Tab(n) terminating in [V ] = λ. Choosing any 0 = vT ∈ VT for each T in Tab(λ), we get a basis {vT | T → λ} of each V λ, called the Gelfand–Tsetlin basis. We note that the choice of vT is uniquely determined only up to a scalar multiple.
ﬁeld means that the Wedderburn–Artin decomposition of An, An = λ∈Irr(n) ε(λ)An ∼ = λ∈Irr(n) End k (V λ), expresses An as a direct sum of matrix algebras over k. In the isomorphism, the central idempotent ε(λ) ∈ An acts as the identity in End k (V λ) and zero in the other components, so {ε(λ) | λ ∈ Irr(n)} is the (unique) partition of unity of the center Z(An).
the subalgebra of An generated by the centers Z(A1), . . . , Z(An). This is the Gelfand–Tsetlin subalgebra of An; it is a commutative subalgebra of An. Deﬁnition (Canonical projections) To each path T : ∅ = λ0 → λ1 → · · · → λn of length n in the branching graph, we associate a unique element εT := ε(λ1)ε(λ2) · · · ε(λn) of the Gelfand–Tsetlin subalgebra Xn. Given an irreducible module V ∼ = V λ for An and any T → λ, the element εT ∈ Xn ⊆ An is the projection mapping V onto VT. These projections are used to prove the following result. Proposition (Okounkov–Vershik) The Gelfand–Tsetlin algebra Xn is the algebra of all elements of An that act diagonally on the Gelfand–Tsetlin basis {vT : T ∈ Tab(λ)} for each irreducible An-module V λ (λ ∈ Irr(n)). In particular, the algebra Xn is a maximal commutative subalgebra of An.
in [Okounkov–Vershik], but it is implicit in the setup. (See also [Goodman–Graber 11].) Proposition Let {An | n ≥ 0} be a MFF. For any n, the set {εT | T ∈ Tab(n)} of canonical projections is a partition of unity in An. It is also a k-basis for the Gelfand–Tsetlin subalgebra Xn. Suppose that we have elements Jk ∈ Xk for all k ∈ N. The sequence of inclusions A1 ⊂ · · · ⊂ An−1 ⊂ An induces a corresponding sequence of inclusions X1 ⊂ · · · ⊂ Xn−1 ⊂ Xn among the Gelfand–Tsetlin subalgebras. So J1, . . . , Jn ∈ Xn. Writing each Jk in terms of the idempotent basis of Xn, we obtain Jk = T∈Tab(n) cT(k) εT, where cT(k) ∈ k. This deﬁnes an n-tuple cT := (cT(1), . . . , cT(n)) ∈ kn for each T ∈ Tab(n) that we call the T-content of the operators J1, . . . , Jn.
a generalized Jucys–Murphy sequence (GJM-sequence) in the MFF {An | n ≥ 0} if Xn = J1, . . . , Jn , for all n ∈ N. If the ﬁeld k is inﬁnite then a GJM-sequence must exist. We can check whether or not a given sequence is a GJM-sequence by means of the content eigenvalues cT = (cT(1), . . . , cT(n)) of J1, . . . , Jn. (Note: cT(k) is the eigenvalue of Jk on the Gelfand–Tsetlin basis element vT.) To be precise, Proposition For a sequence (Jk ∈ Xk)k∈N in a MFF, the following are equivalent. 1 J1, . . . , Jn = Xn. 2 For all S, T ∈ Tab(n), S = T ⇐⇒ cS = cT. This is proved by a “Lagrange interpolation” argument. (Such arguments have appeared before, in work of [Murphy], [Diaconis–Greene], [Ram], etc.)
a path of length n, let T be the path of length n − 1 obtained by omitting its last edge and vertex. Proposition Let (Jk)k∈N be a GJM-sequence in the multiplicity-free family {Ak | k ≥ 0}. Suppose that S, T ∈ Tab(n). Then 1 cT(k) = c T (k) for all k < n. 2 If S = T but S = T then cS(n) = cT(n). This implies that the following “Lagrange interpolation” polynomial is well-deﬁned (see [Garsia] in symmetric groups case): PT(Jn) := S∈Tab(n) S=T, S=T Jn−cS(n) cT(n)−cS(n) . Theorem (Recursion for εT ) If (Jn)n∈N is a GJM-sequence then εT = ε T PT(Jn), for any T ∈ Tab(n).
that zn = J1 + · · · + Jn ∈ Z(An), for all n ∈ N then we call it a Jucys–Murphy sequence (JM-sequence). Such sequences always exist in any MFF, if the ﬁeld k is inﬁnite. Given a path T → λ that terminates in λ we write type(T) = λ. Proposition Assume that (Jn)n∈N is a JM-sequence in a multiplicity-free family. For any T ∈ Tab(n) the polynomial PT(Jn) depends only on type(T), type(T). If λ = type(T) and µ = type(T) then we write Pλ µ (Jn) := PT(Jn). The proposition says this is well-deﬁned. Theorem (Recursion for central idempotents) Assume that (Jn)n∈N is a JM-sequence in a multiplicity-free family. For any λ ∈ Irr(n), we have ε(λ) = µ ε(µ) · Pλ µ (Jn), where µ varies over the set of immediate predecessors of λ in the branching graph B.
The family {εT | T ∈ Tab(n)} is a partition of unity of An and {ε(λ) | λ ∈ Irr(n)} is the partition of unity of Z(An). Knowing one family is equivalent to knowing the other. Let (Jk)k∈N be a JM-sequence. Let Pλ µ (Jk) ∈ Xk be the polynomial deﬁned above, for each edge µ → λ (λ ∈ Irr(k)) in the branching graph B. Then: For any path T ∈ Tab(n), the idempotent εT is the product of all the polynomials along the edges of T. For any λ ∈ Irr(n), the central idempotent ε(λ) is the sum of all such products, taken over all the paths terminating in λ. This extends combinatorial approach to representations from symmetric groups to all MFFs, using paths in the branching graph B in place of tableaux. Remark. JM-sequences are not unique.
{kSn | n ≥ 0} is a MFF. The sequence of JM-elements is deﬁned by setting Jk := (1, k) + (2, k) + · · · + (k − 1, k) (the formal sum of all transpositions moving the point k). Irreps of Sn are indexed by partitions of n, and paths in branching graph by standard tableaux. Young found certain combinatorial bases of the irreps (and kSn itself) called the “seminormal form”. This leads to a family of pairwise orthogonal primitive idempotents that we call the seminormal idempotents. [Thrall 41] gave a recursive formula for the seminormal idempotents (in terms of Young symmetrizers). We show Theorem The seminormal idempotents described by Thrall’s recursion coincide with the canonical idempotents {εT | T a standard tableau}. This was stated without proof in [Murphy 83].
the Brauer algebra Bn(δ) is semisimple (and split) over k provided the parameter δ ∈ k is not an integer [Wenzl 88]. Then {Bn(δ) | n ≥ 0} is an MFF. A sequence of JM-elements [Nazarov 96, etc] is deﬁned by Jk = (1, k) + (2, k) + · · · + (k − 1, k) − (1, k) − (2, k) − · · · − (k − 1, k). Here (i, j) is the pure contraction corresponding to a transposition (i, j). Irreps of Bn(δ) are indexed by partitions of n − 2l, for 0 ≤ 2l ≤ n. Paths T in the branching graph correspond to the up-down tableaux in the literature.