Branching rules for modular representations of symmetric groups. III: Some corollaries and a problem of Mullineux.

*(English)*Zbl 0854.20014Let \(K\) be a field of characteristic \(p>0\), \(\Sigma_n\) the symmetric group on \(n\) letters, \(\Sigma_{n-1}<\Sigma_n\) the subgroup consisting of the permutations of the first \(n-1\) letters, and \(D^\lambda\) the irreducible \(K\Sigma_n\)-module corresponding to a (\(p\)-regular) partition \(\lambda\) of \(n\). In part II [J. Reine Angew. Math. 459, 163-212 (1995; Zbl 0817.20009)] we have described the socle of the restriction \(D^\lambda\downarrow_{\Sigma_{n-1}}\) and obtained a number of other results which can be considered as generalisations of the classical Branching Theorem in zero characteristic. In the present paper we provide some applications of these results.

First, we obtain a lower bound for the dimension of an irreducible module \(D^\lambda\) in terms of the paths in a modular version of the Young graph and (which is essentially the same) in terms of a modular version of the standard \(\lambda\)-tableaux. Then we prove a somewhat surprising result, namely that \(\text{soc}(D^\lambda\downarrow_{\Sigma_{n-1}})\) contains at most \(p\) simple components. Moreover we find explicitly the number of these simple components and prove that all of them belong to distinct blocks. We also obtain some information about indecomposable components of \(D^\lambda\downarrow_{\Sigma_{n-1}}\). Let \[ D^\lambda \downarrow_{\Sigma_{n-1}}=I_1\oplus I_2\oplus\dots\oplus I_d \] be a decomposition into a direct sum of indecomposables. We prove that each of the modules \(I_j\) is self-dual and has simple socle and head (which are isomorphic to each other). Moreover all the \(I_j\)’s belong to distinct blocks and the number \(d\) can be described explicitly. In particular, \(d \leq p\). The use of Frobenius reciprocity allows one to obtain similar results for the induced modules \(D^\lambda \uparrow^{\Sigma_{n+1}}\).

Next we prove that for two irreducibles \(D^\lambda\) and \(D^\mu\) in one block the socles of \(D^\lambda \downarrow_{\Sigma_{n-1}}\) and \(D^\mu \downarrow_{\Sigma_{n-1}}\) “do not intersect”, i.e. do not have common simple components. This result can be developed in the following direction. It is a simple observation that in characteristic zero the socle of the restriction \(S \downarrow_{\Sigma_{n-1}}\) of a simple \(\Sigma_n\)-module \(S\) defines \(S\) uniquely (for \(n \geq 3\)). In general this is not true in characteristic \(p\). However we obtain a technical result describing all the exceptions.

Perhaps the most essential application of the branching rules obtained here concerns a problem of Mullineux. The problem is formulated as follows. Let sgn be the 1-dimensional sign representation of \(\Sigma_n\). It is evident that for any irreducible representation \(D^\lambda\) the tensor product \(D^\lambda\otimes\text{sgn}\) is again irreducible. The question is “which one”? Put \(D^\lambda\otimes\text{sgn}=D^{b(\lambda)}\). Then \(b\) is a bijection on the set \(P_n\) of \(p\)-regular partitions of \(n\), which clearly satisfies the property \(b^2=\text{id}\). G. Mullineux has constructed some bijection, say \(m\), on \(P_n\) and conjectured that \(b=m\).

We push the problem in two directions. First we provide an algorithm different from that of Mullineux, which gives a bijection \(w\) on \(P_n\) and prove that \(b=w\). Thus we obtain the first known description of \(b\). Second we prove that \(b\) is uniquely determined by the following properties. (1) \(\text{core}(b(\lambda))=(\text{core}(\lambda))'\) for all \(p\)-regular \(\lambda\) (where \(\mu'\) denotes the partition conjugate to a partition \(\mu\) and core means the \(p\)-core); (2) for any \(p\)-regular \(\lambda\) there exist a good node \(A\) for \(\lambda\) and a good node \(B\) for \(b(\lambda)\) such that \(b(\lambda\setminus A)=b(\lambda)\setminus B\) (the definition of a good node is introduced in the paper cited above). Mullineux has shown that his map \(m\) enjoys the property (1). So \(m=b\) is equivalent to the property (2) with \(m\) instead of \(b\) in it. Thus we reduce the Mullineux Conjecture to a purely combinatorial question about the Mullineux map \(m\). The above characterisation of \(b\) also seems to be important because of the following reason. If, instead of combinatorial algorithms, one conjectures a simple formula for \(b\), then to verify this formula it will suffice only to check properties (1) and (2).

First, we obtain a lower bound for the dimension of an irreducible module \(D^\lambda\) in terms of the paths in a modular version of the Young graph and (which is essentially the same) in terms of a modular version of the standard \(\lambda\)-tableaux. Then we prove a somewhat surprising result, namely that \(\text{soc}(D^\lambda\downarrow_{\Sigma_{n-1}})\) contains at most \(p\) simple components. Moreover we find explicitly the number of these simple components and prove that all of them belong to distinct blocks. We also obtain some information about indecomposable components of \(D^\lambda\downarrow_{\Sigma_{n-1}}\). Let \[ D^\lambda \downarrow_{\Sigma_{n-1}}=I_1\oplus I_2\oplus\dots\oplus I_d \] be a decomposition into a direct sum of indecomposables. We prove that each of the modules \(I_j\) is self-dual and has simple socle and head (which are isomorphic to each other). Moreover all the \(I_j\)’s belong to distinct blocks and the number \(d\) can be described explicitly. In particular, \(d \leq p\). The use of Frobenius reciprocity allows one to obtain similar results for the induced modules \(D^\lambda \uparrow^{\Sigma_{n+1}}\).

Next we prove that for two irreducibles \(D^\lambda\) and \(D^\mu\) in one block the socles of \(D^\lambda \downarrow_{\Sigma_{n-1}}\) and \(D^\mu \downarrow_{\Sigma_{n-1}}\) “do not intersect”, i.e. do not have common simple components. This result can be developed in the following direction. It is a simple observation that in characteristic zero the socle of the restriction \(S \downarrow_{\Sigma_{n-1}}\) of a simple \(\Sigma_n\)-module \(S\) defines \(S\) uniquely (for \(n \geq 3\)). In general this is not true in characteristic \(p\). However we obtain a technical result describing all the exceptions.

Perhaps the most essential application of the branching rules obtained here concerns a problem of Mullineux. The problem is formulated as follows. Let sgn be the 1-dimensional sign representation of \(\Sigma_n\). It is evident that for any irreducible representation \(D^\lambda\) the tensor product \(D^\lambda\otimes\text{sgn}\) is again irreducible. The question is “which one”? Put \(D^\lambda\otimes\text{sgn}=D^{b(\lambda)}\). Then \(b\) is a bijection on the set \(P_n\) of \(p\)-regular partitions of \(n\), which clearly satisfies the property \(b^2=\text{id}\). G. Mullineux has constructed some bijection, say \(m\), on \(P_n\) and conjectured that \(b=m\).

We push the problem in two directions. First we provide an algorithm different from that of Mullineux, which gives a bijection \(w\) on \(P_n\) and prove that \(b=w\). Thus we obtain the first known description of \(b\). Second we prove that \(b\) is uniquely determined by the following properties. (1) \(\text{core}(b(\lambda))=(\text{core}(\lambda))'\) for all \(p\)-regular \(\lambda\) (where \(\mu'\) denotes the partition conjugate to a partition \(\mu\) and core means the \(p\)-core); (2) for any \(p\)-regular \(\lambda\) there exist a good node \(A\) for \(\lambda\) and a good node \(B\) for \(b(\lambda)\) such that \(b(\lambda\setminus A)=b(\lambda)\setminus B\) (the definition of a good node is introduced in the paper cited above). Mullineux has shown that his map \(m\) enjoys the property (1). So \(m=b\) is equivalent to the property (2) with \(m\) instead of \(b\) in it. Thus we reduce the Mullineux Conjecture to a purely combinatorial question about the Mullineux map \(m\). The above characterisation of \(b\) also seems to be important because of the following reason. If, instead of combinatorial algorithms, one conjectures a simple formula for \(b\), then to verify this formula it will suffice only to check properties (1) and (2).

Reviewer: A.S.Kleshchev (Minsk)

##### MSC:

20C30 | Representations of finite symmetric groups |

05E10 | Combinatorial aspects of representation theory |

20C20 | Modular representations and characters |