On Residue Symbols and the Mullineux Conjecture.
This paper is concerned with properties of the Mullineux map, which plays a rle in p-modular representation theory of symmetric groups. We introduce the residue symbol for a p-regular partitions, a variation of the Mullineux symbol, which makes the detection and removal of good nodes (as introduced by Kleshchev) in the partition easy to describe. Applications of this idea include a short proof of the combinatorial conjecture to which the Mullineux conjecture had been reduced by Kleshchev.
Introduction
It is a well known fact that for a given prime p the p-modular irreducible
representations D - of the symmetric group S n of degree n are labelled in a
canonical way by the p-regular partitions - of n. When the modular irreducible
representation D - of S n is tensored by the sign representation we get a new
modular irreducible representation D - P . The question about the connection
between the p-regular partitions - and - P was answered in 1995 by the proof
of the so-called "Mullineux Conjecture".
The importance of this result lies in the fact that it provides information about
the decomposition numbers of symmetric groups of a completely different kind
than was previously available. Also it is a starting point for investigations
on the modular irreducible representations of the alternating groups. From
a combinatorial point of view the Mullineux map gives a p-analogue of the
conjugation map on partitions. The analysis of its fixed points has led to some
interesting general partition identities [AO], [B].
The origin of this conjecture was a paper by G. Mullineux [M1] from 1979,
when he defined a bijective involutory map - M on the set of p-regular
partitions and conjectured that this map coincides with the map - P . The
statement is the Mullineux conjecture. To each p-regular partition
Mullineux associated a double array of integers, known now as the Mullineux
symbol and the Mullineux map is defined as an operation on these symbols.
The Mullineux symbol may be seen as a p-analogue of the Frobenius symbol
for partitions.
Before the proof of the Mullineux conjecture many pieces of evidence for it had
been found, both of a combinatorial as well as of representation theoretical na-
ture. The breakthrough was a series of papers by A. Kleshchev [K1], [K2],
[K3] on "modular branching", i.e. on the restrictions of modular irreducible
representations from S n to S n\Gamma1 . Using these results Kleshchev [K3] reduced
the Mullineux conjecture to a purely combinatorial statement about the compatibility
of the Mullineux map with the removal of "good nodes" (see below).
A long and complicated proof of this combinatorial statement was then given
in a paper by Ford and Kleshchev [FK].
In his work on modular branching Kleshchev introduced two important notions,
normal and good nodes in p-regular partitions. Their importance has been
stressed even further in recent work of Kleshchev [K4] on modular restriction.
Also these notions occur in the work of Lascoux, Leclerc and Thibon on Hecke
Algebras at roots of unity and crystal bases of quantum affine algebras [LLT]; it
was discovered that Kleshchev's p-good branching graph on p-regular partitions
is exactly the crystal graph of the basic module of the quantized affine Lie
algebra U q ( -
studied by Misra and Miwa [MM].
From the above it is clear that a better understanding of the Mullineux symbols
is desirable including their relation to the existence of good and normal nodes in
the corresponding partition. In the present paper this relation will be explained
explicitly. We introduce a variation of the Mullineux symbol called the residue
symbol for p-regular partitions. In terms of these the detection of good nodes
is easy and the removal of good nodes has a very simple effect on the residue
symbol. In particular this implies a shorter and much more transparent proof
of the combinatorial part of the Mullineux conjecture with additional insights
(Section 4). We also note that the good behaviour of the residue symbols with
respect to removal of good nodes allows to give an alternative description of the
p-good branching graph, and thus of the crystal graph mentioned above. Some
further illustrations of the usefulness of residue symbols are given in Section 3.
This includes combinatorial results on the fixed points of the Mullineux map.
Basic definitions and preliminaries
Let p be a natural number.
Let - be a p-regular partition of n. The p-rim of - is a part of the rim of -
([JK], p. 56), which is composed of p-segments. Each p-segment except possibly
the last contains p points. The first p-segment consists of the first p points of
the rim of -, starting with the longest row. (If the rim contains at most p
points it is the entire rim.) The next segment is obtained by starting in the
row next below the previous p-segment. This process is continued until the
final row is reached. We let a 1 be the number of nodes in the p-rim of - (1)
and let r 1 be the number of rows in -. Removing the p-rim of - (1) we get a
new p-regular partition - (2) of n \Gamma a 1 . We let a be the length of the p-rim
and the number of parts of - (2) respectively. Continuing this way we get a
sequence of partitions -
and a corresponding Mullineux symbol of -
a 1 a 2 \Delta \Delta \Delta am
The integer m is called the length of the symbol. For p ? n, the well-known
Frobenius symbol F (-) of - is obtained from G p (-) as above by
As usual, here the top and bottom line give the arm and leg lengths of the
principal hooks.
It is easy to recover a p-regular partition - from its Mullineux symbol G p (-).
Start with the hook - (m) , given by am backwards. In placing
each p-rim it is convenient to start from below, at row r i . Moreover, by a slight
reformulation of a result in [M1], the entries of G p (-) satisfy (see [AO])
we call the corresponding
column
\Delta of the Mullineux symbol a singular column, otherwise the column
is called regular.
If G p (-) is as above then the Mullineux conjugate - M of - is by definition the
p-regular partition satisfying
a 1 a 2 \Delta \Delta \Delta am
In particular, for p ? n, this is just the ordinary conjugation of partitions.
Example. Let
(In both cases the nodes of the successive 5-rims are numbered 1; 2; 3; 4).
Thus
Now let p be a prime number and consider the modular representations of S n
in characteristic p; note that for all purely combinatorial results the condition
of primeness is not needed.
The modular irreducible representations D - of S n may be labelled by p-regular
partitions - of n, a partition being p-regular if no part is repeated p (or more)
times ([JK], 6.1); this is the labelling we will consider in the sequel.
Tensoring the modular representation D - of S n by the sign representation of
another modular irreducible representation, labelled by a p-regular
partition - P . Mullineux has then conjectured [M1]:
Conjecture. For any p-regular partition - of n we have -
If - is a p-regular partition we let as before
a 1 a 2 \Delta \Delta \Delta am
denote its Mullineux symbol. We then define the Residue symbol R p (-) of -
as
oe
where x j is the residue of am+1\Gammaj \Gamma r m+1\Gammaj modulo p and y j is the residue
of p. Note that the Mullineux symbol G p (-) can be recovered
from the Residue symbol R p (-) because of the strong restrictions on
the entries in the Mullineux symbol. Also, it is very useful to keep in mind
that for a residue symbol there are no restrictions except that
(which would correspond to starting with the p-singular partition (1 p )). We
also note that a column
\Delta in R p (-) is a singular column in G p (-) if and only
and R 5
oe
Also for the residue symbol of a p-regular partition we have a good description
of the residue symbol of its Mullineux conjugate; this is just obtained by
translating the definition of the Mullineux map on the Mullineux symbol to
the residue symbol notation.
Lemma 2.1 Let the residue symbol of the p-regular partition - be
oe
Then the residue symbol of - M is
ae
oe
where
Notation. We now fix a p-regular partition -. Then e
- denotes the partition
obtained from - by removing all those parts which are equal to 1. We will
assume that - has d such parts, 0 - d - - be the
partition obtained from - by subtracting 1 from all its parts. We say that - is
obtained by removing the first column from -. Unless otherwise specified we
assume that the residue symbol R p (-) for - is as above.
For later induction arguments we formulate the connection between the residue
symbols of - and -. First we consider the process of first column removal; this
is an easy consequence of Proposition 1.3 in [BO] and the definition of the
residue symbol.
Lemma 2.2 Suppose that
oe
Then
ae x 0
oe
Here y 0 is defined to be 1 and the - j 's are defined by
Moreover, if x then the first column in R p (-) (consisting of x 0
1 and y 0
omitted.
Remark 2.3 In the notation of Lemma 2.2 the number d of parts equal to 1
in - is determined by the congruence
Moreover since r 1 is the number of parts of - and y is clear that
y m is the p-residue of the lowest node in the first column of -.
Next we consider the relationship between - and - from the point of adding a
column to -; this follows from Proposition 1.6 in [BO].
Lemma 2.4 Suppose that
ae x 0
oe
Then
oe
Here x
and the - 0
j 's are defined by
Moreover, if y 0
then the first column in R p (-) (consisting of
Remark 2.5 In the notation of Lemma 2.2 and Lemma 2.4 we have
Indeed,
(by definition of - j )
(by definition of - 0
3 Mullineux fixed points in a p-block
The p-core - (p) of a partition - is obtained by removing p-hooks as long as
possible; while the removal process is not unique the resulting p-regular partition
is unique as can most easily be seen in the abacus framework introduced
by James. The reader is referred to [JK] or [O1] for a more detailed introduction
into this notion and its properties. We define the weight w of - by
The representation theoretic significance of the p-core is the fact that it determines
the p-block to which an ordinary or modular irreducible character
labelled by - belongs. The weight of a p-block is the common weight of the
partitions labelling the characters in the block.
be a partition of n. Then
is the Young diagram of -, and (i; called a node of -. If
is a node of - and Y (-) n f(i; j)g is again a Young diagram of a partition, then
A is called a removable node and - n A denotes the corresponding partition of
Similarly, if IN is such that Y (-) [ f(i; j)g is the Young
diagram of a partition of n is called an indent node of - and the
corresponding partition is denoted - [ A.
The p-residue of a node A = (i; j) is defined to be the residue modulo p of
denoted res p). The p-residue diagram of - is obtained by writing
the p-residue of each node of the Young diagram of - in the corresponding
place.
The p-content of a partition - is defined by counting the
number of nodes of a given residue in the p-residue diagram of -, i.e. c i is the
number of nodes of - of p-residue i. In the example above, the p-content of -
is
It is important to note that the p-content determines the p-core of a partition.
This can be explained as follows. First, for given
the associated ~n-vector by
for any vector
there is a unique p-core - with this ~n-vector ~n associated to its p-content c(-)
(for short, we also say that ~n is associated to -.) We refer the reader to [GKS]
for the description of the explicit bijection giving this relation. From [GKS]
we also have the following
Proposition 3.1 Let - be a p-core with associated ~n-vector ~n. Then
in i
with
How do we obtain the ~n-vector associated to - from its Mullineux or residue
symbol? This is answered by the following
Proposition 3.2 Let - be a p-regular partition with Mullineux symbol resp.
residue symbol
a 1 a 2 \Delta \Delta \Delta am
and R p
oe
Then the associated ~n-vector
Proof. In the residue symbol, singular columns do not contribute to the n-vector
as they contain the same number of nodes for each residue. So let us
consider a regular column
y
a
r
\Delta in the Mullineux symbol and the corresponding
p-rim in the p-residue diagram. In this case, the contribution only
comes from the last section of the p-rim. The final node is in row r and column
1 so its p-residue is What is the p-residue of the top node
of this rim section? The length of this section is j a (mod p), hence we have
to go j a \Gamma 1 steps from the final node of residue y to the top node of the
section, which hence has p-residue j y
(going a north or east step always increases the p-residue by 1!). Thus going
along the residues in the last section we have a strip
the contribution of the intermediate residues to the ~n-vector cancel out, and
we only have a contribution 1 for n x and \Gamma1 for n y\Gamma1 , which proves the claim. 2
First we use the preceding proposition to give a short proof of a relation already
noticed by Mullineux [M2]:
Corollary 3.3 Let - be a p-regular partition. Then
Proof. Let the residue symbol of - be R p
oe
So by Lemma 2.1 we have R p (- M
ae
oe
with
Now we consider the contributions of the entries in the residue symbol to the
~n-vectors. If x then we get a contribution 1 to
and \Gamma1 to n k (-) on the one hand, and a contribution 1 to n \Gamma(k+1) (- M )
and \Gamma1 to n \Gamma(j+1) (- M ) on the other hand. If x then from column
i in the residue symbol we get neither a contribution to ~n(-) nor to ~n(- M ).
Hence \Gamman \Gamma(j+1) (-) for all j, i.e. if
Now let be the p-content of -, then c(- 0
and hence
(\Gamman
Now we turn to Mullineux fixed points.
Proposition 3.4 Let p be an odd prime and suppose that - is a p-regular
partition with - M . Then the representation D - belongs to a p-block of
even weight w.
Proof. If - M , then its Mullineux symbol is of the form
a 1 a 2 \Delta \Delta \Delta am
where as before " and where a i is even if and
only if pja i .
Now by Proposition 3.1 we have
is the ~n-vector associated to - and ~
1). By Proposition 3.2 we have
For a we do not get a contribution to the ~n-vector. For a i 6j 0
(mod p) with a i \Gamma1j j (mod p) we get a contribution 1 to n j and \Gamma1 to n \Gamma(j+1) .
Note that we can not get any contribution to n p\Gamma1. Thus we have
Now we obtain for the weight modulo 2:
Hence the weight is even, as claimed. 2
For the following theorem we recall the definition of the numbers k(r; s):
is a partition, and
r
In view of the now proved Mullineux conjecture, the following combinatorial
result implies a representation theoretical result in [O2].
Theorem 3.5 Let p be an odd prime. Let - be a symmetric p-core and n 2 IN
with
even. Then
Proof. We set
For - 2 F(-) we consider its Mullineux symbol; as - is a Mullineux fixed point
this has the form
a 1 a 2 \Delta \Delta \Delta am
a 1 +" 1a 2 +"
being even if and only if pja i .
In this special situation the general restrictions on the entries in Mullineux
symbols stated at the beginning of x2 are now given by:
(ii) If a
(iii) If a
(iv) a i is even if and only if pja
(v)
We have already explained before how to read off the p-core of a partition
from its Mullineux symbol by calculating the ~n-vector. In the proof of the
previous proposition we have already noticed that entries a i j 0 (mod p) do
not contribute to the ~n-vector.
Now the partitions (a properties (i) to (iv) above are just
the partitions satisfying the special congruence and difference conditions for
and the congruence set
considered in [B], [AO]. The bijection described there transforms the set of
partitions above into the set
where modN b denotes the smallest positive number congruent to b mod N .
Computing the ~n-vector from the b i 's instead of the a i 's with the formula
given in the previous proof then gives the same answer since the congruence
sequence of the b i 's is the same as the congruence sequence of the regular a i 's.
For a bar partition b 2 D as above we then compute its so called N -bar quo-
tient; since b has no parts congruent to 0 or p modulo N , the bar quotient is a
p\Gamma1-tuple of partitions. For the properties of these objects we refer the reader
to [MY], [O1]. It remains to check that the N-weight of b equals w, i.e. that
the N -bar core
N) of b satisfies
We recall from above that we have for the ~n-vector of -:
and
Hence by Proposition 3.1 we obtain
As remarked before the bijection transforming a
leaves the congruence sequence modulo of the regular elements in a
invariant. Now for determining the N-bar core of b we have to pair off b i 's
congruent to 2j
each only have to know for each such j the number
But this is equal to
Now the contribution to the 2p-bar core from the conjugate runners 2j +1 and
2 is for any value of n j easily checked to be
Thus the total contribution to the 2p-bar core is exactly the same as the one
calculated above, i.e. we have as was to be proved. 2
4 The combinatorial part of the Mullineux conjec-
ture
We are now going to introduce the main combinatorial concepts for our inves-
tigations. The concept of the node signature sequence and the definition of
its good nodes have their origin in Kleshchev's definition of good nodes of a
partition. First we recapitulate his original definition [K2].
We write the given partition in the form
For we then define
a t and fl(i;
Furthermore, for
We then call i normal if and only if for all there exists
and such that
We call i good if it is normal and if fl(i;
Let us translate this into properties of the nodes of - in the Young diagram that
can most easily be read off the p-residue diagram of -. One sees immediately
that fi(i; j) is just the length of the path from the node at the beginning
of the i-th block of - to the node at the end of the jth block of -. The
condition is then equivalent to the equality of the p-residue
of the indent node in the outer corner of the ith block and the p-residue of the
removable node at the inner corner of the jth block.
Similarly, is equivalent to the equality of the p-residues of
the removable nodes at the end of the ith and jth block.
We will say that a node A = (i; j) is above the node below
and write this relation as B % A. Then a removable node A of - is
normal if for any B 2 indent node of - above A with res
res Ag we can choose a removable node CB of - with A % CB % B and res
res A, such that jfC j. A node A is good if it is the
lowest normal node of its p-residue.
Consider the example 5. In the p-residue diagram
below we have included the indent node at the beginning of the second block,
marked 3, and we have also marked the removable node of residue 3 at the end
of the fourth block in boldface. The equality of these residues corresponds to
We also see immediately from the diagram below that
4 0The set M i corresponds in this picture to taking the removable node, say A, at
the end of the ith block and then collecting into M i (resp. MA ) all the indent
nodes above this block of the same p-residue as A. For i resp. A being normal,
we then have to check whether for any such indent node, B say, at the end of
the jth block we can find a removable node C = CB between A and B of the
same p-residue, and such that the collection of all these removable nodes has
the same size as M i (resp. MA ). The node A (resp. i) is then good if A is the
lowest normal node of its p-residue.
The critical condition for the normality of i resp. A above is just a lattice con-
dition: it says that in any section above A there are at least as many removable
nodes of the p-residue of A as there are indent nodes of the same residue.
With these notions the Mullineux conjecture was reduced by Kleshchev to the
combinatorial
Conjecture. Let - be a p-regular partition, A a good node of -. Then there
exists a good node B of the Mullineux image - M such that (-nA)
Now we define signature sequences.
A (p)-signature is a pair c" where c 2 f0;
is a sign. Thus 2+ and 3\Gamma are examples of 5-signatures.
A (p)-signature sequence X is a sequence
where each c i " i is a signature.
Given such a signature sequence X we define for
We make the conventions that an empty sum is 0 and that + is counted as +1
in the sum.
The i'th peak value - i (X) for X is defined as
and the i'th end value defined as
We call i a good residue for X if - In that case let
and we then say that the residue c k at step k is i-good for X, for short: c k is
i-good for X. Let us note that if c k is i-good for X then c
Indeed, if is clear since otherwise -
contrary to the definition of c k .
contrary to the
definition of - i (X).
The residue c l is called i-normal if c l is i-good for the truncated sequence
The following is quite obvious from the definitions.
Lemma 4.1 Let X  be a signature sequence and let
X be obtained from X   by adding a signature c t " t at the end. For
the following statements are equivalent
c t is i-good for X.
We are going to define two signature sequences based on -, the node sequence
N(-) and the Mullineux sequence M(-). Although they are defined in very
different ways we will show that they have the same peak and end value for
each i.
The node sequence N(-) consists of the residues of the indent and removable
nodes of -, read from right to left, top to bottom in -. For each indent residue
the sign is + and for each removable residue the sign is \Gamma.
Let us note that according to Remark 2.3 the final signature in N(-) is
Example. Let we have only indicated
the removable and indent nodes in the 5-residue diagram of -.4
Residue
Peak value
Good
(The good signatures (peaks) are underlined and the normal signatures marked
with a prime.)
In other words, in the node sequence N(-) defined before, if c corresponds
to the removable node A, then c and A is normal if and only
if the sequence of signs to the left of A belonging to c j 's with c res A is
latticed read from right to left. Again, the node A resp. c m is good if it is the
last normal node of its residue resp. of its value. The peak value of the node
sequence N(-) is the number of normal nodes of -.
Remark 4.2 Let as before denote e - the partition obtained from - by removing
all those parts which are equal to 1, and let - be the partition obtained from -
by subtracting 1 from all its parts. From the definitions it is obvious that for
all i
Proposition 4.3 Let - and - be as above, and let d be the number of parts 1
in -.
and if
(3) We have
unless the following conditions are all fulfilled
In that case y m is i-good for N(-) and
Proof. Assume that N(-) consists of m 0 signatures (m 0 is odd). Then
N(-) consists of
Suppose that
If
then
where in both sequences c m \Gamma. From this and the definition
of end values (1) and (2) follows easily. Also since the final sign is \Gamma we have
proving (3) in this case.
Suppose d 6= 0.
If again
Again (1) and (2) follows easily. To prove (3) we consider the sequence
Obviously
for all i. The final signature of N(-) has no influence on - i (N(-)), since the
sign is \Gamma. Therefore in order for - i (N   (-)) to be different from - i (N(-)) we
need 4.1. Thus condition (i) of
(3) is fulfilled and condition (iii) follows from ( ). Since by assumption d 6= 0
(ii) is also fulfilled. Thus (3) is proved in this case also. 2
We proceed to prove an analogue of Proposition 4.3 for the Mullineux (signa-
ture) sequence M(-), which is defined as follows:
Let the residue symbol of - be
oe
Then
Starting with the signature 0\Gamma corresponds to starting with an empty partition
at the beginning which just has the indent node (1; 1) of residue 0.
oe
and
(The good signatures in M(-) are again underlined and the normal signatures
marked with a prime.)
The table above is identical with the one in the previous example.
Lemma 4.4 Let - and - be as above. Let M   (-) be the signature sequence
obtained from M(-) by removing the two final signatures y m+ and (y
Then for all i we have
Proof. We use the notation of Lemma 2.2 for R p (-) and R p (-) and proceed
by induction on m. First we study the beginnings of M   (-) and M(-). We
compare
(from M   (-))
with
We have put brackets [ ] around a part of (2), because these signatures do
not occur when x
The former gives a contribution \Gamma1 to residue 1 and contributions 0 to all
others, the latter a contribution \Gamma1 to residue 0 and 0 to all others.
The signatures 0\Gamma 0+ in the latter sequence have no influence on the end
values and peak values of M(-), (even when x may be ignored.
Then again we see that (1) gives the same contribution to residue i as (2) 0 to
residue our result is true.
We assume that the result is true for partitions whose Mullineux symbols have
length and have to compare
(from M   (-))
with
(from M(-))
By Lemma 2.2, (4) may be written as
We see that up to rearrangement the difference between the residues occurring
in (3) and (4) 0 is just ffi m . Whereas the rearrangement is irrelevant for the end
values it could influence the peak value if signatures with same residue but
different signs are interchanged. The possible coincidences of residues with
different signs are
(first and fourth residue in (3))
or
(second and third residue in (3))
But the equations (ff) and (fi) are equivalent and by Lemma 2.2 they are
fulfilled if and only if
becomes
and
In this case the difference between the occurring residues in 1 (without rear-
rangement) and our statement is true.
If y then the difference between the occurring
residues is again 1(= since there is no coincidence for residues with
different signs we may apply Lemma 4.1 and the induction hypotheses to prove
the statement in this case too. 2
Lemma 4.5 Suppose that in the notation as above we have for
Then d 6= 0.
Proof. Suppose
ends on (i \Gamma 1)\Gamma. But then clearly !
Lemma 4.6 Let the notation be as in Lemma 4.4.
(1) For
y j is i-good for M   (-)
ae
j+1 is (i \Gamma 1)-good for M(-)
or
j+1 is (i \Gamma 1)-good for M(-)
(2) For
x j is i-good for M   (-)
j is (i \Gamma 1)-good for M(-)
Proof. This follows immediately from the proof of Lemma 4.4. It should be
noted that x 1 cannot be 0-good for M   (-) since M   (-) starts by 0\Gamma. More-
over, the proof of Lemma 4.4 shows that if x j is i-good for M   (-), then we
cannot have
Proposition 4.7 Let - and - be as above.
and if
(3) We have
In that case y m is i-good
for M(-) and
Note. There is a strict analogy between the Propositions 4.3 and 4.7. In part
(3) the assumption d 6= 0 is not necessary in Proposition 4.7 due to Lemma 4.5.
Proof. By Lemma 4.4
If we add y m+ and (y m \Gamma 1)\Gamma to M   (-) we get M(-). Therefore an argument
completely analogous to the one used in the case d 6= 0 in the proof of Proposition
4.3 may be applied. 2
Theorem 4.8 Let - be a p-regular partition.
Then for all
Proof. We use induction on the number ' of columns in -. For
d
and R p
ae 0
oe
. Thus
and the result is clear. Assume the result has been proved for partitions with
2. Let - be obtained by removing the first column from -.
By the induction hypothesis we have
for all i. Using Proposition 4.3 and Proposition 4.7 (see also the note to Proposition
4.7) we get the result. 2
Theorem 4.9 The following statements are equivalent for a p-regular partition
(1) There is a good node of residue i in -.
(2) M(-) has i as a good residue.
(3) N(-) has i as a good residue.
Proof. (1) , (3): See the beginning of this section.
(2) , (3): Theorem 4.8. 2
Finally we describe the effect of the removal of a good node on the residue
(or equivalently on the Mullineux symbol). First we prove a lemma:
Lemma 4.10 Suppose that there is a good node of residue i in -. Then the
following statements are equivalent:
(1) The good node of residue i occurs in the first column of -.
Proof. The statement (1) clearly is equivalent to
(where as before e - is obtained from - by removing all parts equal to 1.) We
now have
(by Theorem 4.8)
is i-good for M(-) (by Proposition 4.7)Theorem 4.11 Suppose that the p-regular partition - has a good node A of
residue i. Let
oe
Then for some j, 1 - j - m, one of the following occurs:
(1) x j is i-good for M(-) and
oe
(2) y j is i-good for M(-) and
oe
resp.
oe
Proof. The proof is by induction on j-j. Suppose first that A occurs in the
first column of -. Then the first column in G p (-nA) is obtained from the
first column in G p (-) by subtracting 1 in each entry and all other entries are
unchanged; note that in the case where G p
, we have a degenerate
case and G p (-nA) is the empty symbol. By definition of the residue symbol
this means that y m in R p (-) is replaced by y of course, in
the degenerate case also R p (-nA) is the empty residue symbol. On the other
hand y m is i-good for M(-) by Lemma 4.10, and in the degenerate case y 1 is
0-good for M(-), and so we are done in this case.
Now we assume that A does not occur in the first column of -. Let B be the
node of - corresponding to A. Clearly B is a good node of residue
We may apply the induction hypothesis to - and B. Suppose that
ae x 0
oe
By the induction hypothesis we know that one of the following cases occurs:
Case I. x 0
j in R p (-) is replaced by x 0
j is (i \Gamma 1)-good
for M(-).
Case II. y 0
j in R p (-) is replaced by y 0
j is (i \Gamma 1)-good
for M(-), resp. in the degenerate case y 0
1 is 0-good for M(-) and then the first
column x 0y 0in R p (-) is omitted in R p (-nB).
We treat Case I in detail. Case II is treated in a similar way.
Case I: By Lemma 4.6 we have one of the following cases:
We add a first column to -nB to get -nA. Then R p (-nA) is obtained from
using Lemma 2.4. We fix the notation
ae x 00
oe
and
oe
Case Ia. We know x 0
since we are in Case I and y since we are
in Case Ia. Moreover since
Also x 0
2.2. By Lemma 2.4
where
ae 0 if x 00
But x 00
by the above.
Thus
1. It is readily seen that all other entries
in R p (-nA) coincide with those of R p (-). Thus possibility (2) occurs in the
theorem.
Case Ib. We know x 0
since we are in Case I and x since we are
in Case Ib. Also since
i.e. x 00
2. Let ffi 00
again be defined by
ae 0 if x 00
Then by Lemma 2.4 x
. We claim that ffi 00
j and we also know that x 0
by definition of M(-) x 0
j is not a peak, contrary to our assumption that we are
in Case I. Thus ffi 00
Again it is easily seen that
all other entries of R p (-nA) coincide with those of R p (-). Thus possibility (1)
occurs in the theorem. 2
We illustrate the theorem above by giving Kleshchev's p-good branching graph
for p-regular partitions for in both the usual and the
residue symbol notation; we recall that the p-good branching graph for p-
regular partitions is also the crystal graph for the basic representation of the
quantum affine algebra (see [LLT] for these connections).
Below, an edge from a partition - of m to a partition - of labelled
by the residue r if - is obtained from - by removing a node of residue
those edges are drawn that correspond to the removal of good nodes.
@
@ @
@
@ @
@
@ @
@
@ @
@
@ @
@
@ @
@
@ @
@
@ @
We can now easily deduce the combinatorial conjecture to which the Mullineux
conjecture had been reduced by Kleshchev:
Corollary 4.12 Suppose that the p-regular partition - has a good node A of
residue i. Then its Mullineux conjugate - M has a good node B of residue \Gammai
satisfying
Proof. Considering the residue symbol of - it is easily seen that the Mullineux
sequence of - and its conjugate - M are very closely related. Indeed, the peak
and end values for each residue i in M(-) equal the corresponding values for
the residue \Gammai in M(- M ), and if there is an i-good node at column k in the
residue symbol of -, then there is an \Gammai-good node at column k in the residue
symbol of - M . More precisely, in the regular case these good nodes are one
at the top and one at the bottom of the column, whereas in the singular case
both are at the top. Comparing this with Theorem 4.11 implies the result. 2

Acknowledgements

. The authors gratefully acknowledge the support of the
Danish Natural Science Foundation and of the EC via the European Network
'Algebraic Combinatorics' (grant ERBCHRXCT930400).



--R

Partition identities with an application to group representation theory
A combinatorial proof of a refinement of the Andrews- Olsson partition identity
Theory (A) 68
A proof of the Mullineux conjecture
Cranks and t-cores
The representation theory of the symmetric group
Branching rules for modular representations of symmetric groups I
Branching rules for modular representations of symmetric groups II
Branching rules for modular representations of symmetric groups III
On decomposition numbers and branching coefficients for symmetric and special linear groups
Hecke algebras at roots of unity and crystal basis of quantum affine algebras
Crystal base of the basic representation of U q
Some combinatorial results involving Young diagrams
Bijections of p-regular partitions and p-modular irreducibles of symmetric groups
On the p-cores of p-regular diagrams
Combinatorics and representations of finite groups
The number of modular characters in certain blocks
--TR

--CTR
Jonathan Brundan , Jonathan Kujawa, A New Proof of the Mullineux Conjecture, Journal of Algebraic Combinatorics: An International Journal, v.18 n.1, p.13-39, July
signature sequence;good nodes in residue diagram;modular representation;mullineux conjecture;symmetric group