TRANSITIVE POWERS OF YOUNG-JUCYS-MURPHY ELEMENTS ARE CENTRAL
I.P. GOULDEN∗ AND D.M. JACKSON†
ABSTRACT. Although powers of the Young-Jucys-Murphy elements Xi = (1 i) + (2 i) + · · · + (i −
1 i), i = 1, . . . , n, in the symmetric group Sn acting on {1, . . . , n} do not lie in the centre of the
group algebra of Sn, we show that transitive powers, namely the sum of the contributions from
elements that act transitively on [n], are central. We determine the coefficients, which we call star
factorization numbers, that occur in the resolution of transitive powers with respect to the class basis
of the centre of Sn, and show that they have a polynomiality property. These centrality and poly-
nomiality properties have seemingly unrelated consequences. First, they answer a question raised
by Pak [P] about reduced decompositions; second, they explain and extend the beautiful symmetry
result discovered by Irving and Rattan [IR]; and thirdly, we relate the polynomiality to an existing
polynomiality result for a class of double Hurwitz numbers associated with branched covers of the
sphere, which therefore suggests that there may be an ELSV-type formula (see [ELSV]) associated
with the star factorization numbers.
1. INTRODUCTION AND BACKGROUND
We begin with an account of the main theorem of this paper and its relationship to the enumer-
ation of a class of ramified covers of the sphere, a question that arises in algebraic geometry.
1.1. Young-Jucys-Murphy elements and the Main Theorem. The Young-Jucys-Murphy elements
in the group algebra CSn of the symmetric group Sn on [n] := {1, . . . , n}, are given by
Xi = (1 i) + (2 i) + · · ·+ (i− 1 i), i = 1, . . . , n,
where X1 = 0 (see, e.g., [VO] for a detailed description and further references). Let Z(n) denote the
centre of CSn, n ≥ 1. Then the algebra generated by Z(1), . . . , Z(n) is called the Gel’fand-Tsetlin
algebra, and one of the key results described in [VO] is the fact that this algebra is also generated
by X1, . . . ,Xn, despite the fact that Xn is clearly not contained in Z(n) for any n > 2.
We define a linear operator T on CSn by T(σ1 · · · σr) = σ1 · · · σr, if the group generated by
the permutations σ1, · · · , σr acts transitively on [n], and T(σ1 · · · σr) = 0 otherwise. The subject
of this paper is T Xrn, for an arbitrary non-zero integer r, which we call a transitive power of Xn.
It is straightforward matter to apply T to Xrn =
i∈[n−1](i n)
, since the only products not
annihilated are those containing at least one occurrence of (i n) as a factor for every i ∈ [n − 1].
It follows immediately from the Principle of Inclusion-Exclusion that a transitive power can be
written explicitly as
T Xrn =
γ⊆[n−1]
(−1)|γ|Xn(γ)
where Xn(γ) =
j∈γ(j n).
Date: April 5, 2007.
∗Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada.;
ipgoulden@uwaterloo.ca.
†Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada.;
dmjackso@math.uwaterloo.ca.
http://arxiv.org/abs/0704.1100v1
2 I. P. GOULDEN AND D. M. JACKSON
Our main result, Theorem 1.1, is that the transitive powers of Xn, unlike powers, are contained
in Z(n). Moreover, since a basis for Z(n) is given by the set of all Kα where Kα =
π, and Kα
is the conjugacy class of Sn (naturally) indexed by the partition α of n, then Theorem 1.1 expresses
T Xrn as an explicit linear combination of the Kα.
We use the following notation and terminology for partitions. If α1, · · · , αk are positive integers
with 1 ≤ α1 ≤ · · · ≤ αk and α1 + · · · + αk = n, then α = (α1, . . . , αk) is a partition of n with k
parts, and we write α ⊢ n and l(α) = k, for n, k ≥ 0. Let α \ αj denote the partition obtained by
removing the single part αj from α, for any j = 1, . . . , k. Let α ∪m denote the partition obtained
by inserting a single new part equal to m into α (placed in the appropriate ordered position). Let
2α = (2α2, . . . , 2αk), and aα = aα1 · · · aαk for any indeterminates a1, a2, . . .. Let P denote the set of
all partitions, including the empty partition ε, which it a partition of 0 with 0 parts. If α has fj parts
equal to j for each j ≥ 1, then we also use (1f12f2 · · · ) to denote α, and we write |Aut α| =
j≥1 fj!
In the statement of our main result we use the notation pi ≡ pi(α) to denote the i-th power sum
of the parts of the partition α, i ≥ 1, and qi ≡ qi(α) := pi+ p1− 2, i ≥ 2, and we define ξ2j and ξ by
2j := log (ξ(x)) , where ξ(x) := 2x−1sh
where sh and ch denote, respectively, hyperbolic sine and cosine.
Theorem 1.1 (Main Theorem). For r ≥ 0, TXrn is contained in the centre Z(n) of CSn. Moreover, the
resolution of TXrn with respect to the class basis of Z(n) is
TXrn =
ag(α)Kα
where the range of summation on the right hand side is restricted by the condition n + m − 2 + 2g = r,
with m = l(α), and ag(α) is a polynomial in the parts of α given by
ag(α) =
(n+m− 2 + 2g)!α1 · · ·αmQg(α), where Qg :=
ξ2βq2β
|Aut β|
, g ≥ 0.
For example, the explicit expressions for small genera g = 0, . . . , 5 are
Q0 = 1, Q1 =
q2, Q2 =
(−2q4 + 5q22) , Q3 =
(16q6 − 42q4 2 + 35q23) ,
3 · 2710!
(−144q8 + 320q6 2 + 84q42 − 420q4 22 + 175q24) ,
3 · 2812!
(768q10 − 1584q8 2 − 704q6 4 + 1760q6 22 + 924q42 2 − 1540q4 23 + 385q25) ,
1.2. Background.
1.2.1. Minimal factorizations into star transpositions. We now turn our attention temporarily to a
another point of view. The transpositions (1 a), for a = 2, . . . , n, are called star transpositions
in Sn, with the distinguished element 1 (it appears in each transposition) referred to as the pivot
element. An ordered factorization (τ1, . . . , τr) of σ ∈ Sn into star transpositions is said to be
transitive if the group generated by τ1, . . . , τr acts transitively on [n]. For a transitive factorization
of σ ∈ Kα into r star transpositions, a result in [GJ0] implies that r = n + m − 2 + 2g for some
non-negative integer g, where α has m parts. Thus r ≥ n + m − 2, and we refer to transitive
factorizations into n+m− 2 star transpositions as minimal.
Pak [P] enumerated minimal factorizations (he called them reduced decompositions) into star
transpositions for permutations fixing the pivot element 1, with exactly m other cycles, each of
TRANSITIVE POWERS OF YOUNG-JUCYS-MURPHY ELEMENTS ARE CENTRAL 3
length k ≥ 2. More recently, Irving and Rattan [IR] generalized Pak’s result by considering min-
imal factorizations of arbitrary permutations into star transpositions, and proved the following
elegant result.
Theorem 1.2 ([IR]). For each permutation σ ∈ Kα with α = (α1, . . . , αm), α1+· · ·+αm = n and m,n ≥
1, the number of transitive factorizations of σ into n+m− 2 star transpositions is
(n +m− 2)!
α1 · · ·αm.
Because of the apparent asymmetry of these factorizations (i.e., the pivot element 1 appears in
every factor), the fact that Theorem 1.2 is constant on conjugacy classes is particularly surprising
(we shall refer to this fact as the centrality property of Theorem 1.2). The proofs given in [P] and [IR]
are bijective, involving restricted words and plane trees.
In terms of factorizations into star transpositions, the number ag(α) given by the Main Theorem
clearly can be interpreted as the number of transitive factorizations of each σ ∈ Kα into n +m −
2 + 2g star transpositions, with pivot element n. We shall call ag(α) a star factorization number.
Thus Theorem 1.2 is precisely the case g = 0 of Theorem 1.1 (the necessary relabelling of the pivot
element is justified by the centrality of these results). The investigation described in this paper
answers Pak’s [P] question about an explicit expression for the general case. It was motivated by
Irving and Rattan’s paper, in our attempt to determine whether the centrality of their remarkable
result for star factorizations with a minimum number of factors persisted for star factorizations
with an arbitrary number of factors.
1.2.2. Connections with algebraic geometry. The connection to algebraic geometry is made through
Hurwitz’s encoding [H] of an n-sheeted branched cover of the sphere in terms of transpositions
that represent the sheet transitions at the elementary branch points. In this context, the transi-
tivity of the factorizations corresponds to the connectedness of the cover. From this perspective,
the coefficient ag(α) in Theorem 1.1 counts genus g branched covers of the sphere in which the
branching over the point 0 is specified by α, and there are n + m − 2 + 2g other simple branch
points, each of which corresponds to a transition between sheet number n (the pivot sheet) and an-
other sheet. For the corresponding transitive factorizations into star transpositions, we therefore
also refer to g as the genus of the factorization (e.g., Theorem 1.2 counts genus 0 factorizations). For
further details about branched covers, see, for example, [GJ0], [GJVn], [GJV] and [H].
The double Hurwitz number H
(n),α
is equal to the number of genus g branched covers of the
sphere in which the branching over the points 0 and ∞ is specified by (n) and α, respectively,
together with m− 1 + 2g other simple branch points. A scaling of this double Hurwitz number to
bg(α) := α1 · · ·αmH
(n),α
gives the number of transitive factorizations of each σ ∈ Kα into m − 1 + 2g transpositions and
a single n-cycle. There is a striking similarity between Theorem 1.1 and the following result, in
which the notation q̂i := pi − 1, i ≥ 1 is used.
Theorem 1.3 ([GJV]). For r ≥ 0, the resolution of Kr
(1n−2 2)
Kn with respect to the class basis of the centre
Z(n) of CSn is
(1n−2 2)Kn =
bg(α)Kα,
4 I. P. GOULDEN AND D. M. JACKSON
where the range of summation on the right hand side is restricted by the condition m − 1 + 2g = r, with
m = l(α), and bg(α) is a polynomial in the parts of α given by
bg(α) = (m− 1 + 2g)!n
m−2+2g α1 · · ·αm Q̂g(α), where Q̂g :=
ξ2β q̂2β
|Aut β|
, g ≥ 0.
This is a restatement of Theorem 3.1 in [GJV], which gives a formula for the double Hurwitz
number H
(n),α
, since Kr
1n−22
Kn = T(K
1n−22
Kn) (each term in Kn acts transitively on [n]).
1.2.3. Two relationships between Theorems 1.1 and 1.3. To explore a more direct relationship between
Theorems 1.1 and 1.3, we now give two expressions for ag(α) in terms of the bh(γ)’s.
The first is very simple and expresses ag(α), which enumerates factorizations in Sn, directly in
terms of bg(α ∪ 1
n−1), which enumerates factorizations in S2n−1.
Corollary 1.4. For g ≥ 0 and α a partition of n with m parts, we have
ag(α) =
n!(2n− 1)n+m−3+2g
bg(α ∪ 1
n−1).
Proof. In the notation of Theorems 1.1 and 1.3, clearly qi(α) = q̂i(α∪1
n−1), so Qg(α) = Q̂g(α∪1
n−1).
The result follows immediately from Theorems 1.1 and 1.3. �
The second expresses ag(α) as a linear combination of bg−h(α), 0 ≤ h ≤ g, each of which
enumerates factorizations in Sn.
Corollary 1.5. For g ≥ 0 and α, a partition of n with m parts, we have
ag(α) =
bg−h(α)
nm−2+2g−2h
n+m− 2 + 2g
n− 1 + 2h
) n−1∑
(−1)j
(n− 1)− j
)n−1+2h
Proof. In the notation of Theorems 1.1 and 1.3, clearly qi(α) = q̂i(α)+n−1. Then from Theorems 1.1
and 1.3, and (1), we have
Qg(α)x
2g = exp
ξ2jq2j(α)x
 = ξ(x)n−1
Q̂g(α)x
But, for h ≥ 0, we have (using the notation [A]B to denote the coefficient of A in B)
[x2h]ξ(x)n−1 = [xn−1+2h]
2 − e−
(−1)j
(n− 1)− j
)n−1+2h
(n − 1 + 2h)!
and, together with (2), this gives
Qg(α) =
Q̂g−h(α)
(n− 1 + 2h)!
(−1)j
(n − 1)− j
)n−1+2h
The result follows immediately from Theorems 1.1 and 1.3. �
TRANSITIVE POWERS OF YOUNG-JUCYS-MURPHY ELEMENTS ARE CENTRAL 5
1.3. Outline. In Section 2, we introduce a generating series for the number of transitive factoriza-
tions into star transpositions in arbitrary genus, and prove that it is the unique formal power series
solution of a linear partial differential equation that we call the Join-cut Equation for this class of
factorizations. The proof is based on a join-cut analysis of these factorizations, since the left-most
factor σ either joins two cycles of the product π of the remaining factors to form one cycle or cuts
one cycle of π into two, depending on whether the two elements moved by σ are, respectively, in
different cycles of π, or in the same cycle. This approach has been applied previously where the
factors are arbitrary transpositions, for the genus 0 case in [GJ0], and for arbitrary genus in [GJVn]
and [GJV].
In Section 3, we solve the Join-cut Equation to obtain the generating series for transitive factor-
izations into star transpositions in arbitrary genus. Then, by determining the coefficients in this
generating series, we prove Theorem 1.1 (and hence also give a new proof of Theorem 1.2).
In Section 4, we pose some questions that arise from this investigation, but that we have been
unable to resolve.
2. THE JOIN-CUT EQUATION
Let SA denote the symmetric group on an arbitrary set A. For an arbitrary set A of size n
containing 1 (for convenience, we shall consider star transpositions with pivot element 1), let K
denote the set of all permutations in SA in which 1 lies on a cycle of length i and the remaining
cycle-lengths in the disjoint cycle representation are given by the parts of α, where α ⊢ n − i, for
n ≥ i ≥ 1. It is straightforward to determine, independently of the choice of A, that
(3) |K(i)α | =
(i− 1)!|Kα| =
(n− 1)!
α1 · · ·αk|Aut α|
where α = (α1, . . . , αk). Consider a fixed permutation σ ∈ K
α in Sn, and let cg(i, α) be the
number of transitive factorizations of σ into n + k − 1 + 2g star transpositions (this number is
constant for each such σ because of the symmetry of elements 2, . . . , n; note that σ lies in the
conjugacy class Kα∪i, which has m = k + 1 cycles). Let Ψ denote the generating series
(4) Ψ(t, u, x; z,y) :=
n≥i≥1,
k,g≥0
un+k−1+2g
(n+ k − 1 + 2g)!
x2gzi
α⊢n−i,
l(α)=k
|K(i)α |cg(i, α)yα.
The following result is the Join-cut Equation for the set of transitive factorizations into star trans-
positions. It states that Ψ is annihilated by the partial differential operator
(5) ∆ :=
i,j≥1
∂zi+j
i,j≥1
jzi+j
∂zi∂yj
Theorem 2.1 (Join-cut Equation). The generating series Ψ = Ψ(t, u, x; z,y) is the unique formal power
series solution of ∆Ψ = 0, with initial condition Ψ(t, 0, x; z,y) = z1t.
Proof. Fix a triple (k, g, i) of integers with k, g ≥ 0 and i ≥ 1 to be other than (0, 0, 1). Also fix
a partition α with l(α) = k and a permutation σ ∈ K
α in Sn, where α ⊢ n − i. Consider a
transitive factorization (τ1, . . . , τr) of σ into star transpositions, where r = n+ k − 1 + 2g. For this
factorization, we let π = τ2 · · · τr = τ1σ, and τ1 = (1 a). There are cg(i, α) such factorizations of
σ, and we obtain a recurrence equation for cg(i, α) by considering the following case analysis for
these factorizations which is based on the left-most factor τ1.
6 I. P. GOULDEN AND D. M. JACKSON
Case 1: τ1 6= τj for any j = 2, . . . , r. In this case, the element a is a fixed point in π, and (τ2, . . . , τr)
is not a transitive factorization of π. But, if we let π′ ∈ S[n]\{a}, whose disjoint cycle representation
is obtained by removing the one-cycle containing a from the disjoint cycles of π, then (τ2, . . . , τr)
is a transitive factorization of π′. But σ is obtained from π′ by inserting a immediately before 1
in the cycle of π′ containing 1. This implies that π′ ∈ K
(i−1)
α in S[n]\{a}. Note that the transitive
factorization (τ2, . . . , τr) of π
′ has r− 1 = (n− 1) + k− 1 + 2g factors and that this is reversible, so
we conclude that the number of such factorizations is cg(i− 1, α), the contribution from this case.
Case 2: τ1 = τj for some j = 2, . . . , r. In this case, (τ2, . . . , τr) is a transitive factorization of π, since
for a product of star transpositions in Sn to be transitive, it is necessary and sufficient that each of
(1 2), . . . , (1n) appears at least once as a factor (as observed in [IR]). There are two subcases, based
on which disjoint cycles of π contain elements 1 and a.
Subcase 2(a): 1 and a appear on the same cycle of π. In this subcase, that cycle of π is cut into
two cycles in σ, one containing 1, and the other containing a. Consequently, for each factorization
(τ1, . . . , τr) of σ, we obtain a factorization of π in this subcase by selecting a to be any element
on the k cycles of σ not containing 1. We account for the choices of a on these cycles as follows:
Suppose the cycles are indexed so they have lengths α1, . . . , αk (the cycles are all non-empty, so
they are distinguishable, even if their lengths are equal). If a is on the jth such cycle, of length
αj , then there are αj choices of a, and the cycle of π containing 1 has length i + αj , so we have
π ∈ K
(i+αj )
in Sn. Since the transitive factorization (τ2, . . . , τr) of π has r− 1 = n+(k− 1)− 1+2g
factors and this is reversible, we conclude that there are cg(i+αj , α\αj) such factorizations, giving
a total contribution from this subcase of
j=1 αjcg(i+ αj , α \ αj).
Subcase 2(b): 1 and a appear on different cycles of π. In this subcase, these cycles of π
are joined into a single cycle of σ, containing both 1 and a. Consequently, for each factorization
(τ1, . . . , τr) of σ, we obtain a factorization of π in this subcase by selecting a to be any other ele-
ment on the cycle of σ containing 1. We account for these i − 1 choices of a as follows: Suppose
that the cycle of σ containing 1, in cyclic order, is (1 ji−1 . . . j1) (i.e., so σ(1) = ji−1, σ(jt) = jt−1, for
t = 2, . . . , i − 1, and σ(j1) = 1). If a = jm, then π has disjoint cycles (1 ji−1 . . . jm+1) (containing
1) and (jm . . . j1), together with all the cycles of σ not containing 1, so we have π ∈ K
(i−m)
α∪m in Sn,
and the transitive factorization (τ2, . . . , τr) of π has r− 1 = n+ (k+1)− 1+ 2(g− 1) factors. Since
this is reversible, we conclude that there are cg−1(i −m,α ∪m) such factorizations, giving a total
contribution from this subcase of
m=1 cg−1(i−m,α ∪m).
Adding together the contributions from these disjoint cases, we obtain the linear recurrence
equation
cg(i, α) = cg(i− 1, α) +
αjcg(i+ αj , α \ αj) +
cg−1(i−m,α ∪m),
for k, g ≥ 0, i ≥ 1 (except for the simultaneous choices k = g = 0 and i = 1) and α with
l(α) = k. The partial differential equation follows by multiplying this recurrence equation by
ntn u
n+k−2+2g
(n+k−2+2g)!
xgzi|K
α |yα, and summing over the above range of k, g, i, α.
The initial condition follows from the fact that there is a single, empty factorization with no
factors, of the single permutation (with 1 as a fixed point) in S1. Thus we have c0(1, ε) = 1. �
TRANSITIVE POWERS OF YOUNG-JUCYS-MURPHY ELEMENTS ARE CENTRAL 7
3. A PROOF OF THEOREM 1.1
3.1. An explicit solution to the Join-cut Equation. The next result gives the explicit solution of
the Join-cut Equation in terms of the series ξ defined in (1) and W ≡ W (t, u, x : z) where
zℓξ(ℓux)ξ(ux)
ℓ−2uℓ−1tℓ.
Theorem 3.1. Let Z := t ∂
W (t, u, x : z) and Y := ξ(ux)2u2W (t, u, x : y). Then Ψ = ZeY .
Proof. It is a straightforward matter to show that the Join-cut Equation with the given boundary
condition has a unique solution. The remainder of the proof is a verification that ∆ annihilates Ψ
and that the boundary condition is satisfied.
The operator ∆ is a linear combination of four differential operators. It is straightforward to
obtain the four expressions for the application of each of these operators to Ψ. Let x̂ := ux for
brevity. Then the expressions are:
ℓzℓξ(x̂)
ℓ−3uℓ−2tℓ
(ℓ− 1)ξ(ℓx̂)ξ(x̂) + ℓx̂ξ′(ℓx̂)ξ(x̂) + (ℓ− 2)x̂ξ(ℓx̂)ξ′(x̂)
ymξ(x̂)
m−1umtm
(m+ 1)ξ(mx̂)ξ(x̂) +mx̂ξ′(mx̂)ξ(x̂) +mx̂ξ(mx̂)ξ′(x̂)
e−Y t
izi+1ξ(ix̂)ξ(x̂)
i−2ui−1ti+1
i+ 1 +
mymξ(mx̂)ξ(x̂)
mum+1tm
i,j≥1
∂zi+j
i,j≥1
(i+ j)ziyjξ
(i+ j)x̂
ξ(x̂)i+j−2ui+j−1ti+j,
e−Y x2
i,j≥1
jzi+j
∂ziyj
i,j≥1
ijzi+jξ(ix̂)ξ(jx̂)ξ(x̂)
i+j−2ui+jti+j
zℓξ(x̂)
ℓ−2uℓ−2tℓSℓ,(6)
where, with r := exp
, we have
i,j≥1,
i+j=ℓ
(ri − r−i)(rj − r−j) = (ℓ− 1)(rℓ + r−ℓ)− 2
rℓ−1 − r−ℓ+1
r − r−1
Now let θ := 1
x̂, and substituting this expression for Sℓ in (6), we obtain the revised fourth ex-
pression
e−Y x2
i,j≥1
jzi+j
∂ziyj
zℓξ(x̂)
ℓ−2uℓ−2tℓ
(ℓ− 1)ch(ℓθ)−
(ℓ− 1)θ
sh(θ)
Combining these four expressions, and recalling the definition (5) of the partial differential opera-
tor ∆, we have
(7) e−Y ∆Φ =
zℓξ(x̂)
ℓ−3uℓ−2tℓTℓ +
ℓ,m≥1
zℓymξ(x̂)
ℓ+m−3uℓ+m−1tℓ+mUℓ,m,
where Tℓ, for ℓ ≥ 1 and Uℓ,m, for ℓ,m ≥ 1, are explicit polynomials in hyperbolic cosines and
hyperbolic sines of multiples of θ, and in θ, using (1). It is readily shown, using the addition
formulae for hyperbolic sines and cosines that Tℓ = 0 for ℓ ≥ 1 and, similarly, that Uℓ,m = 0 for
8 I. P. GOULDEN AND D. M. JACKSON
ℓ,m ≥ 1. Thus, from (7), we have ∆Φ = 0. But ξ(0) = 1, so Φ|u=0 = z1t and we conclude from
Theorem 2.1 and the uniqueness of the solution of the Join-cut Equation that Ψ = Φ, giving the
result. �
3.2. An expression for the coefficients of Ψ. It is now straightforward to determine the coeffi-
cients in the generating series Ψ, and thus obtain a proof of Theorem 1.1.
Proof of Theorem 1.1. Suppose that α is a partition of n − i with k parts. Then for all n ≥ i ≥ 1,
k, g ≥ 0, Theorem 3.1 and (1) gives ([A]B denotes the coefficient of A in B)
[tnun+k−1+2gx2gziyα]Ψ =
|Aut α|
[u2gx2g]ξ(iux)ξ(ux)i−2
l(α)∏
ξ(αjux)ξ(ux)
|Aut α|
[x2g]ξ(x)n−2ξ(ix)
l(α)∏
ξ(αjx)
|Aut α|
[x2g] exp
ξ2jq2j(α ∪ i)x
so, together with (3) and (4), this gives
cg(i, α) =
(n+ k − 1 + 2g)!
α1 · · ·αk i
ξ2βq2β(α ∪ i)
|Aut β|
But this is symmetric in α1, . . . , αk, i, and the result follows immediately by renaming α ∪ i as α,
which has m = k + 1 parts. �
4. FURTHER QUESTIONS
The following questions arise in the light of the results of this paper:
(1) Is it possible to find a simple proof of the centrality in Theorem 1.1, without evaluating the
class coefficients ag(α)? This might follow from a decomposition for Young-Jucys-Murphy
elements, or from a more elementary argument in the symmetric group.
(2) Is it possible to give a direct proof of Corollary 1.4 or 1.5 – i.e., to establish these relation-
ships between ag(α) and bg(α) without appealing, as we have, to the explicit formulae?
This would be particularly interesting, since bg(α), as defined, is clearly central. Such a
proof might involve Young-Jucys-Murphy elements, or a more elementary argument in
the symmetric group, or the geometry of branched covers. Presumably such a proof would
contain a solution to Question 1 above.
(3) In [GJV], the polynomiality of bg(α) (in the parts of α) in Theorem 1.3, was the basis for a
conjectured ELSV-type formula for H
(n),α
, involving a Hodge integral over some, unspec-
ified, moduli space. Does the polynomiality of ag(α) in Theorem 1.1 also lead to a similar
ELSV-type formula when ag(α) is rescaled as a covering number?
ACKNOWLEDGEMENTS
The work of both authors was supported by Discovery Grants from NSERC. We thank John
Irving, Igor Pak, Amarpreet Rattan and Ravi Vakil for helpful comments.
TRANSITIVE POWERS OF YOUNG-JUCYS-MURPHY ELEMENTS ARE CENTRAL 9
REFERENCES
[ELSV] T. Ekedahl, S. Lando, M. Shapiro and A. Vainstein, Hurwitz numbers and intersections on moduli spaces of curves,
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[GJ0] I. P. Goulden and D. M. Jackson, Transitive factorizations into transpositions and holomorphic mappings on the sphere,
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[GJ1] I. P. Goulden and D. M. Jackson, A proof of a conjecture for the number of ramified coverings of the sphere by the torus,
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[GJ3] I. P. Goulden and D. M. Jackson, The number of ramified coverings of the sphere by the double torus, and a general
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http://arxiv.org/abs/math/0610640
http://arxiv.org/abs/math/0503040
	1. Introduction and background
	1.1. Young-Jucys-Murphy elements and the Main Theorem
	1.2. Background
	1.3. Outline
	2. The Join-cut Equation
	3. A proof of Theorem ??
	3.1. An explicit solution to the Join-cut Equation
	3.2. An expression for the coefficients of 
	4. Further questions
	Acknowledgements
	References

  Although powers of the Young-Jucys-Murphya elements X_i = (1 i) + ... +(i-1
i), i = 1, ..., n, in the symmetric group S_n acting on {1, ...,n} do not lie
in the centre of the group algebra of S_n, we show that transitive powers,
namely the sum of the contributions from elements that act transitively on {1,
>...,n}, are central. We determine the coefficients, which we call star
factorization numbers, that occur in the resolution of transitive powers with
respect to the class basis of the centre of S_n, and show that they have a
polynomiality property. These centrality and polynomiality properties have
seemingly unrelated consequences. First, they answer a question raised by Pak
about reduced decompositions; second, they explain and extend the beautiful
symmetry result discovered by Irving and Rattan; and thirdly, we relate the
polynomiality to an existing polynomiality result for a class of double Hurwitz
numbers associated with branched covers of the sphere, which therefore suggests
that there may be an ELSV-type formula associated with the star factorization
numbers.

Introduction and background
	1.1. Young-Jucys-Murphy elements and the Main Theorem
	1.2. Background
	1.3. Outline
	2. The Join-cut Equation
	3. A proof of Theorem ??
	3.1. An explicit solution to the Join-cut Equation
	3.2. An expression for the coefficients of 
	4. Further questions
	Acknowledgements
	References
