Generating functions for fixed points of the Mullineux map

Let $S_{n}$ denote the symmetric group on $n$ letters. Recall that a partition $\lambda=(\lambda_{1},\lambda_{2},\ldots,\lambda_{s})\vdash n$ is $e$-regular if no part repeats $e$ or more times. Let $k$ be an algebraically closed field of characteristic $p$. The irreducible $kS_{n}$ modules are labelled by $p$-regular partitions [Jam78, Chapter 11] and are denoted $\{D^{\lambda}\mid\lambda\textrm{ is p-regular}\}$. Define an involution $P$ on the $p$-regular partitions by $D^{\lambda}\otimes\operatorname{sgn}\cong D^{P(\lambda)}.$ In [Mul79], Mullineux defined a combinatorial map $\lambda\rightarrow m_{e}(\lambda)$ on the set of $e$-regular partitions. He conjectured for $e=p$ a prime that $m_{p}=P$. Almost twenty years later, Kleshchev [Kle96] described $P$ and Ford and Kleshchev [FK97] proved the conjecture by proving Kleshchev’s description matched Mullineux’s. The bijection $m_{e}$ is defined combinatorially on $e$-regular partitions for $e$ arbitrary, and can be interpreted similarly using irreducible representations of a certain Hecke algebra [Mat99, Chapter 6.3]. For $e=2$ the sign representation is trivial and the Mullineux map is the identity. Combinatorial properties of this map have inspired much research, often independent of representation theoretic applications.

Let $A_{n}$ be the alternating group and let the characteristic of $k$ be an odd prime $p$. One can count irreducible $kA_{n}$ modules in two ways: group theoretically by counting $p$-regular conjugacy classes of $A_{n}$, or by restricting modules from $S_{n}$ using Clifford theory and counting in terms of fixed points of the map $P$. In 1991 Andrews and Olsson, using Olsson’s work in [Ols92], counted the fixed points of $m_{p}$:

This answer agreed with the known representation theoretic count for fixed points of $P$, providing evidence for the as-yet-unproven Mullineux conjecture by showing $m_{p}$ had the expected number of fixed points. Later Bessenrodt and Olsson refined this by computing fixed points in an arbitrary $p$-block of weight w:

Our main results are to extend Theorems 1.1 and 1.2 to arbitrary $e$. For $e$ odd but not prime it is a simple observation that the original proofs carry over, while for $e$ even additional work is needed. The author would like to acknowledge his colleague William Keith for useful discussions about generating functions.

For a partition $\lambda=(\lambda_{1},\lambda_{2},\ldots,\lambda_{s})$ with Young diagram $[\lambda]$, define the rim of $\lambda$ to be the boxes along the southeast edge of the diagram, i.e. boxes $(i,j)\in[\lambda]$ with $(i+1,j+1)\not\in[\lambda]$. Now consider a subset of the rim defined as follows, and called the $e$-rim. Starting at the top right of the rim, take the first $e$ elements on the rim. Then move to the rightmost element of the rim in the next row, and take the next $e$ elements. Continue until the final row is reached, observing that the final segment may contain fewer than $e$ boxes.

For example if $\lambda=(7,7,7,4,4,1,1)\vdash 31$ and $e=5$ we have the e-rim:

Let $a_{1}$ be the number of boxes and $r_{1}$ be the number of rows in the $e$-rim, so in our example $a_{1}=12$, $r_{1}=7$.

To define the Mullineux symbol $G_{e}(\lambda)$, remove the $e$-rim, and then calculate the $e$-rim of what remains to determine $(a_{2},r_{2})$. Continue this process until all boxes are removed. Assemble these numbers in an array, called the Mullineux symbol of $\lambda$:

Notice that $(a_{1},a_{2},\ldots,a_{k})$ is also a partition of the same integer $n$. For example from Figure 2.2 we see that:

Now define $\epsilon_{i}=0$ if $e\mid a_{i}$ and $\epsilon_{i}=1$ otherwise, and set $s_{i}=a_{i}-r_{i}+\epsilon_{i}.$ Then we have Mullineux’s conjecture (now theorem):

It is easy to reconstruct $\lambda$ from $G_{e}(\lambda)$ so Proposition 2.1 gives a combinatorial description of the Mullineux map. For example with $\lambda$ as in Figure 2.2

and $m_{5}(7,7,7,4,4,1,1)=(12,9,4,2,2,2).$ Thus a fixed point of $m_{e}$ will have a Mullineux symbol of the form:

where $a_{i}$ is even if and only if $e\mid a_{i}$ if and only if $\epsilon_{i}=0$.

In his original paper Mullineux described necessary and sufficient conditions for such an array to arise as the Mullineux symbox of an $e$-regular partition:

Now we can apply Proposition 2.2 to arrays of the form (2.2), i.e. fixed points. The conditions on the $r_{i}$ in Proposition 2.2 can easily be translated to conditions on the $a_{i}$. Thus we can enumerate fixed points simply by counting the suitable partitions $(a_{1},a_{2},\ldots,a_{k})\vdash n$.

Then we have:

The proof of Proposition 2.4 goes through for $e$ arbitrary, so to enumerate Mullineux fixed points we need to understand the set $\mathcal{M}_{e}(n)$. The criteria defining $\mathcal{M}_{e}(n)$ are not easily translated into a generating function. The main theorem (Theorem 2) of [AO91] gives a remarkable enumeration of partitions with difference conditions like this. Observe for $p$ odd that $\mathcal{M}_{p}(n)$ is a set of type $P_{2}(A;N,n)$ from that paper, where $N=2p$ and $A=\{1,3,5,\ldots p-2,p+2,\ldots 2p-1\}$. The paper gives a bijection with a set $P_{1}(A;2p,n)$, which in this case is just partitions with distinct odd parts not divisible by $p$, giving Theorem 1.1.

In Table 1 we list the number of fixed points under the Mullineux map when $e=4$ for $1\leq n\leq 20$ and $e$-weights $0\leq w\leq 5$. Weight zero partitions are $e$-cores and $m_{e}$ acts as conjugation on $e$-cores, so the first column of this table enumerates self-conjugate 4-core partitions, which is sequence A053692 in the Online Encyclopedia of Integer Sequence (OEIS) [OEI]. Each successive column is shifted down by $e=4$ and multiplied by the corresponding entry in the sequence $\{1,1,3,4,9,12,23\ldots\}$, which is A002513 in the OEIS, counting partitions of $n$ with even parts of two colors, also known as “cubic partitions”. Our main results generalize Theorems 1.1 and 1.2 to arbitrary $e$ and explain the structure of this table.

We fix some standard generating function notation. Define the $q$-Pochhammer symbol:

and the Ramanujan $\chi$ function:

Observe that $\chi(q)$ is the generating function counting partitions of $n$ into distinct odd parts.

Let $mf_{e}(n)$ be the number of $e$-regular partitions of $n$ fixed by the Mullineux map and let

be the corresponding generating function. Our first result determines this generating function:

Notice that when $e=2$ that $MF_{2}(q)$ simplifies to $\prod_{k\text{ odd}}\frac{1}{1-q^{k}}$, the generating function counting partitions with odd parts, which is known to be the same as that for distinct parts. Here, the Mullineux map is trivial and we obtain $mf_{2}(n)$ is the number of two-regular partitions, i.e. partitions with distinct parts, as expected.

If we look at the corresponding alternating series, there is a nice common description of the two generating functions:

For $e=3,4,5,6$ the generating function $MF_{e}(-q)$ corresponds to the sequences A098884, A261734, A133563 and A261736 respectively in the OEIS [OEI]. As of this writing there is no mention of the Mullineux map in any of these entries!

Theorems 1.1 and 1.2 are stated for $p$ an odd prime, because that is where the original representation theory motivation comes from. However there is nothing in either proof that makes use of primality, so the $e$ odd case of Theorems 3.1 and 3.7 should be attributed to those authors. Thus we will consider only the case where $e$ is even.

When $e$ is odd notice that Definition 2.3(i) gives that if $e\mid a_{i}$ then actually $N=2e\mid a_{i}$. This is not the case for $e$-even, and this means that $\mathcal{M}_{e}(n)$ in this case is not dealt with by the bijections in [AO91]. However in [Bes95], Bessenrodt gave a vast generalization that includes this case. So in her notation let $N=2e$ for $e$ even. Choose the sets $A^{\prime}=\{e\}$ and $A^{\prime\prime}=\{1,3,5,\ldots,2e-1\}$ with $A=A^{\prime}\cup A^{\prime\prime}.$ One can check that the conditions for $\mathcal{M}_{e}(n)$ are precisely those defining Bessenrodt’s set $P_{2}(A^{\prime},A^{\prime\prime};N,n).$ Bessenrodt gives a bijection with a set $P_{1}(A^{\prime},A^{\prime\prime};N,n)$, which are all partitions with parts congruent mod $2e$ to elements of $A$ and repeating parts must be congruent to elements of $A^{\prime}$. This is precisely the set of partitions with all parts odd or odd multiples of $e$ with the odd parts distinct. This proves Theorem 3.1(b).

Finally, to prove Theorem 3.7(b), we need to count Mullineux fixed points in a given block. We will follow the idea of [BO98, Theorem 3.5], although the proof is slightly more complicated since $e$ is even.

Consider the generating function in (3.6). The first term is counting $e/2$ tuples of partitions with even parts and the second term counts partitions with odd parts. We can equally well count $e/2$ tuples of arbitrary partitions but then double all the parts. So Theorem 4.2 immediately implies Theorem 3.7(b), and all that remains is to prove Theorem 4.2.

For the next result we will need the following observation, which is easy to see in [Bes95] (recalling that $N=2e$.)

Thus we can use the $b_{i}$’s to calculate $\vec{n}$ and the corresponding $e$-core and $e$-weight as in Section 2.1. Recall that Bessenrodt’s $P_{1}(A^{\prime},A^{\prime\prime};N,n)$ in this case are partitions of $n$ with all parts either odd or odd multiples of $e$, and the odd parts must be distinct. Henceforth we will consider these as pairs of partitions $\{(c_{1},c_{2},\ldots,c_{k}),e\mu\}$ where $(c_{1},c_{2},\ldots,c_{k})$ has distinct odd parts and $\mu$ has odd parts. Notice that the set of nonzero residues mod $e$ in $(c_{1},c_{2},\ldots,c_{k})$ is the same as in $(b_{1},b_{2},\ldots b_{t})$, since the odd multiples of $e$ all have $e$-residue zero To complete the proof we will need to use the theory of $t$-bar cores and $t$-bar quotients.