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In representation theory, a very popular set of finite dimensional algebras are the $q$-Schur algebras, which are given by looking at the endomorphisms of $V^{\otimes d}$ where $V$ is the standard representation of the quantum group $U_q(\mathfrak{gl}_n)$. One should think of this as kind of enhanced version of the Hecke algebra for the symmetric group that doesn't lose simple representations at roots of unity.

The cyclotomic $q$-Schur algebra is a generalization of this, where the symmetric group is replaced the complex reflection group $S_n\wr C_\ell$ (this is the group of monomial matrices, where one allows $\ell$th roots of unity as coefficients). For more details, see the original paper of Dipper, James and Mathas.

The original definition is as the endomorphism ring of a collection of modules over the Ariki-Koike algebra called "permutation modules" $M^\nu$. These are generalizations (and deformations) of the permutation representations of $S_n$ on subsets of $n$ elements, and are in bijection with $\ell$-tuples $\nu$ of partitions with $n$ total boxes.

This gives a natural collection of projectives $N_\nu=\oplus_{\mu}\mathrm{Hom}(M^\nu,M^\mu)$, which generate the category of representations, but are very far from being irreducible. On the other hand , the indecomposable projectives $P_\mu$ of the cyclotomic $q$-Schur algebra are also bijection with these $\ell$-tuples of partitions.

So, given a collection of multipartitions, one can ask which indecomposable projectives occur in $N_\nu$ for these multipartitions; I'd be interested to know any good references for this problem, as I only know the fairly obvious things about it (i.e. what you can deduce from the generic case, etc.). However, there is one set I'm particularly interested in.

My question: If I consider only multipartitions where each constituent partition is $(1^a)$ for some $a$ (or whatever corresponds to the regular rep of $S_n$ in your conventions), which projectives appear in $N_\nu$?

My conjecture: The projectives corresponding to multipartitions where all constituent partitions are $k$-restricted, where $k$ is the order of $q$ in the complex numbers.

My follow-up questions: What if I only consider multipartitions of the form $(1^a)$ and require that for some $S\subset [1,\ell]$ the corresponding partitions are empty?

My follow-up conjecture: The projectives where for each block of consecutive elements in $S$ followed by an element not in $S$, the multipartition for that piece is $k$-Kleshchev.

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