Generalized hook lengths in symbols and partitions

Abstract

In this paper, we present, for any integer d, a description of the set of
hooks in a d-symbol. We then introduce generalized hook length functions for a
d-symbol, and prove a general result about them, involving the core and
quotient of the symbol. We list some applications, for example to the
well-known hook lengths in integer partitions. This leads in particular to a
generalization of a relative hook formula for the degree of characters of the
symmetric group discovered by G. Malle and G. Navarro in [3].

Full text

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arXiv:1101.5067v1 [math.CO] 26 Jan 2011
GENERALIZED HOOK LENGTHS IN SYMBOLS AND
PARTITIONS
CHRISTINE BESSENRODT, JEAN-BAPTISTE GRAMAIN, AND JØRN B. OLSSON
Abstract. In this paper, we present, for any integer d, a description
of the set of hooks in a d-symbol. We then introduce generalized hook
length functions for a d-symbol, and prove a general result about them,
involving the core and quotient of the symbol. We list some applications,
for example to the well-known hook lengths in integer partitions. This
leads in particular to a generalization of a relative hook formula for the
degree of characters of the symmetric group discovered by G. Malle and
G. Navarro in [3].
In his work [2] on unipotent degrees in reflection groups G. Malle used
d-symbols as labels and defined hooks in d-symbols and associated a length
to a hook. With these he was able to prove a “hook formula” for the degrees.
In this paper we introduce generalized hook length functions for d-symbols
and prove a general result about them. More specifically we consider certain
functions h from the set H(S) of hooks of a d-symbol S to R and decom-
positions of the multiset H(S) of all generalized hook lengths h(z), where
z ∈ H(S). For a given positive integer ℓ we find a decomposition which is
compatible with the ℓ-core and ℓ-quotient of S.
Based on a crucial well controlled correspondence between hooks in sym-
bols and hooks in associated core and quotient symbols, our main result
(Theorem 3.2) on generalized hook lengths for symbols is deduced. This
may seem quite special, but in view of its consequence, Theorem 3.3, it has
a number of applications. For instance, we show that the relative hook for-
mula obtained by Malle and Navarro [3, Theorem 9.1] is just the well-known
hook formula for the degree of the irreducible characters of the symmetric
groups with the hooks suitably arranged (Remark 4.12). If H(λ) is the mul-
tiset of hook lengths for a partition λ, and λ has d-core partition λ(d)then
we have in particular H(λ(d)) ⊂ H(λ). Furthermore the remaining elements
Date: January 25, 2011.
2000 Mathematics Subject Classification 20C30, 20C33, 05A17
Keywords: Symbols, Hooks, Hook Lengths, Partitions, Core, Quotient
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2CHRISTINE BESSENRODT, JEAN-BAPTISTE GRAMAIN, AND JØRN B. OLSSON
of H(λ) may be seen as modified hook lengths of a d-quotient partition for λ
(Theorem 4.7).
Our paper is organized as follows. In section 1 we explain the relations
between partitions, β-sets and d-symbols, their hooks and their correspond-
ing cores and quotients. In section 2 we set up useful bijections between the
set of hooks of a symbol and the set of hooks of its core and quotient. This
is used in section 3 to prove our key result, Theorem 3.3. The theorem is
then applied to partitions in section 4 and to symbols in section 5.
1. On β-sets and d-symbols
A β-set X is a finite subset of N0. For s ∈ N0 we put X+s= (X +
s) ∪ {0,1,...,s − 1}. Write X = {a1,a2,...,at} where we always assume
a1> a2> ··· > at. Then we associate to the β-set X the partition p(X)
having the non-zero numbers among ai− (t − i), i = 1,...,t, as parts. For
example, p({5,3,0}) = (3,2). Note that p(X) = p(X+s) for all s ∈ N0.
Let d ∈ N. We define a d-symbol S = (X0,X1,...,Xd−1) as a d-tuple of
β-sets. In analogy with β-sets we define S+s= (X+s
0,X+s
1,...,X+s
d−1).
In [2], a d-symbol is an equivalence class of such d-tuples. The equivalence
relation is generated by the following operations:
• cyclic permutation of the β-sets
• replacing S by S+1.
The multiset of all hook lengths in S, as defined in [2], only depends on
the equivalence class of S. The more general definition of hook lengths in
symbols which we introduce in section 3 includes those of [2], but they are
not independent of the cyclic permutations of the β-sets.
Let s ∈ N. We set [s] = {0,1,... ,s − 1}. For n ∈ N0, we define n[s]∈ [s]
as the remainder of n after division by s.
Definition 1.1. Given d ∈ N we associate to a β-set X a d-symbol
sd(X) = (X(d)
0,X(d)
1,...,X(d)
d−1)
where for j ∈ [d]
X(d)
j
= {k ∈ N0| j + kd ∈ X} .
Clearly, we have
Lemma 1.2. The map sdis a bijection between the set of all β-sets and the
set of all d-symbols.
Definition 1.3. To an arbitrary d-symbol S, we associate a partition p(S),
defined by
p(S) = p(s−1
d(S)).

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GENERALIZED HOOK LENGTHS IN SYMBOLS AND PARTITIONS3
Definition 1.4. (1) A hook in the β-set X is a pair (a,b) of nonnegative
integers with a > b such that a ∈ X and b / ∈ X. The set of hooks in X is
denoted H(X). If a − b = ℓ then (a,b) is called an ℓ-hook in X.
(2) A hook in the d-symbol S = (X0,X1,...,Xd−1) is a quadruple (a,b,i,j)
of nonnegative integers where a ≥ b, i,j ∈ [d], a ∈ Xi, b / ∈ Xj, and in addi-
tion if a = b then i > j. If a = b, we call the hook short and otherwise long.
The set of hooks in S is denoted by H(S). If a − b = ℓ and (i − j)[d]= e,
e ∈ [d] then (a,b,i,j) is called an (ℓ,e)-hook in S.
Note. (i) To avoid confusion we want to point out that since we will be
dealing with generalized hook lengths below, an ℓ-hook will not be the same
as a hook of length ℓ.
(ii) In [2] only long hooks are considered in symbols, since short hooks
give only trivial contributions to the unipotent degrees. The inclusion of
short hooks makes our later arguments much simpler, and the short hooks
will then be dealt with separately whenever necessary.
Lemma 1.5. (1) There is a canonical bijection hX between the set H(X)
of hooks in a β-set X and the set H(p(X)) of hooks in the partition p(X).
Thereby an ℓ-hook in H(X) is mapped to an ℓ-hook in the partition H(p(X)).
(2) There is a canonical bijection hX,dbetween the set H(X) of hooks in
a β-set X and the set H(S) of hooks in the associated d-symbol S = sd(X).
(3) There is a canonical bijection hSbetween the set H(S) of hooks in a
d-symbol S and the set H(p(S)) of hooks in the partition p(S).
Proof. Let X = {a1,a2,...,at}. (1) If (a,b) ∈ H(X), a−b = ℓ and a = ai,
then the partition p(X) has an ℓ-hook in the ith row. The map hXsending
(a,b) to this hook is a bijection; this is well known, see [1, Section 2.7] or [4,
section 1] for more details.
(2) The map hX,dwhich maps (a,b) ∈ H(X) to (a′,b′,i,j) ∈ H(S) where
a = a′d+i, b = b′d+j with i,j ∈ [d], is obviously a bijection between H(X)
and H(S) = H(sd(X)).
(3) By Definition 1.3, p(S) = p(s−1
X = s−1
d(S).
d(S)). Put hS = hX◦ h−1
X,d, where
?
Definition 1.6. (1) Let X be a β-set and z = (a,b) ∈ H(X). If X′=
(X \{a})∪{b} we say that X′is obtained by removing the hook z from X.
(2) Let S = (X0,...,Xd−1) be a d-symbol and z = (a,b,i,j) ∈ H(S).
If i ?= j, we set X′
(Xi\ {a}) ∪ {b}, and we set X′
k= Xkfor all k ?= i,j. Then we say that
S′= (X′
d−1) is obtained by removing the hook z from S.
i= Xi\ {a},X′
j= Xj∪ {b}, for i = j we set X′
i=
0,...,X′

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4CHRISTINE BESSENRODT, JEAN-BAPTISTE GRAMAIN, AND JØRN B. OLSSON
Remark 1.7. If we keep removing ℓ-hooks from a β-set X for a fixed ℓ
we eventually reach a β-set with no ℓ-hooks left. This is the ℓ-core of X,
denoted Cℓ(X). By an abacus argument this is well-defined [1, 2.7.16]. A
similar statement is true for (ℓ,e)-hooks in a symbol [2, 3.4]. We return to
this in Section 5.
We call a d-symbol S = (X0,...,Xd−1) balanced if |X0| = |X1| = ··· =
|Xd−1|, and if in addition there is an i ∈ [d] such that 0 / ∈ Xi.
To a d-tuple (κ0,κ1,...,κd−1) of partitions we associate a balanced d-
symbol as follows. If r is the maximal length (i.e., number of parts) of the
partitions κi, we may choose a β-set Yiof cardinality r for each partition κi.
Then 0 / ∈ Yjwhenever κjhas length r because then Yjis just the set of first
column hook lengths of κj. Thus
td(κ0,κ1,...,κd−1) = (Y0,Y1,...,Yd−1)
is a well-defined balanced d-symbol for the d-tuple of partitions. On the
other hand, if S = (X0,...,Xd−1) is balanced, then td(p(X0),...,p(Xd−1)) =
(X0,...,Xd−1).
We have shown:
Lemma 1.8. There is a bijection tdbetween the set of d-tuples of partitions
and the set of balanced d-symbols.
Definition 1.9. (1) The balanced quotient Q(S) of an arbitrary d-symbol
S = (X0,X1,...,Xd−1) is defined as the balanced d-symbol
Q(S) = td(p(X0),p(X1),...,p(Xd−1)).
We call q(S) = p(Q(S)) the quotient partition of S.
(2) The core C(S) of an arbitrary d-symbol S = (X0,X1,...,Xd−1) is de-
fined as the d-symbol
C(S) = ([x0],[x1],...,[xd−1]),
where xi= |Xi| for i ∈ [d]. We call c(S) = p(C(S)) the core partition of S.
Remark 1.10. A remark on notation. The core C(S) of a d-symbol S
is really its (1,0)-core (see Remark 1.7) and we will consider Q(S) as the
(1,0)-quotient of S. This will be generalized in Section 5.
Remark 1.11. We may recover a d-symbol S = (X0,X1,...,Xd−1) from its
balanced quotient Q(S) = (Y0,Y1,...,Yd−1) together with its core C(S) =
(Z0,Z1,...,Zd−1). Indeed, for i ∈ [d] Ximust be the β-set of cardinality
|Zi| for the partition p(Yi).

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GENERALIZED HOOK LENGTHS IN SYMBOLS AND PARTITIONS5
Definition 1.12. Let X be a β-set. Then the d-quotient partition of X
is defined as qd(X) = q(sd(X)) and the d-core partition of X is defined as
cd(X) = c(sd(X)).
With notation as in Definition 1.1 we define the d-quotient of X as
Qd(X) = s−1
d(td(p(X(d)
0),p(X(d)
1),...,p(X(d)
d−1))) .
Remark 1.13. If we put the elements of X on the d-abacus and if X(d)
as in Definition 1.1 then the results of [1, Section 2.7] or [4, section 1] show
the following
• The d-core partition cd(X) of X is also the d-core λ(d)of the partition
λ = p(X) and thus it stays the same when we replace X by X+s.
• We have cd(X) = p(Cd(X)).
• The d-quotient partition qd(X) of X must have an empty d-core. It
may change if we replace X by X+1. However X and X+dhave the same
d-quotient partition.
• We have qd(X) = p(Qd(X)) and sd(Qd(X)) = Q(sd(X)).
• If t ∈ [d] is chosen such that |X| + t is divisible by d then
j
is
(p(X(d)
t ),p(X(d)
t+1)...,p(X(d)
t+d−1))
is the d-quotient of the partition p(X) [1, 2.7.29]. Here the subscripts are to
be read modulo d.
Lemma 1.14. Let X be a β-set, d ∈ N. Then
|p(X)| = |qd(X)| + |cd(X)|.
Proof. Using Remark 1.13 we get
|p(X)| = d(|p(X(d)
0)| + |p(X(d)
1)| + ··· + |p(X(d)
d−1)|) + |cd(X)|.
If we apply this result to the d-quotient of X,
Qd(X) = s−1
d(td(p(X(d)
0),p(X(d)
1),...,p(X(d)
d−1))),
and the partition qd(X) = p(Qd(X)) (which has an empty d-core) we get
|qd(X)| = |p(Qd(X))| = d(|p(X(d)
follows.
0)|+|p(X(d)
1)|+...+|p(X(d)
d−1)|). The result
?
The following diagram illustrates the connection between a β-set X, its
associated d-symbol S = sd(X), their cores and quotients and the associated
partitions via the partition map p: