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The $p$-blocks of the symmetric group $S_n$ are labelled by pairs $(\gamma, w)$ where $\gamma$ is a $p$-core partition, $w \in \mathbb{N}_0$ is the weight of the block, and $|\gamma| + p w = n$. Scopes proved that, given a weight $w$, there exists $N_p(w)$ such that any $p$-block of a symmetric group is Morita equivalent to a $p$-block of a symmetric group $S_n$ with $n \le N_p(w)$. Therefore there exists $D_p(w)$ such that all the entries in the decomposition matrix of a $p$-block of a symmetric group of weight $w$ are at most $D_p(w)$, with equality attained at least once.

For example, all decomposition numbers in a block of weight $0$ or $1$ are either $0$ or $1$, so $D_p(0) = D_p(1) = 1$. It follows easily from Scopes' bound that $D_2(2) = 2$ and it is known by work of Richards that if $p$ is odd then $D_p(2) = 1$.

What is the best known upper bound on $D_p(w)$?

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    $\begingroup$ Work of Fayers implies that $D_p(3)=1$ and $D_p(4)=2$ if $p>3$. You can extract the exact values of $D_p(3)$ and $D_p(4)$ for small primes from the literature. In general, decomposition numbers can be arbitrarily large: you can see this already in the RoCK blocks, where the decomposition numbers are known explicitly as sums of products of Littlewood-Richardson coefficients. It should be possible to use the RoCK blocks to give an upper bound for $D_p(w)$, when $w>p$. The obvious conjecture is that $D_p(w)\le w$, but I have no real evidence for this. $\endgroup$
    – Andrew
    Commented May 8, 2021 at 9:52
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    $\begingroup$ References: 1. Fayers, "Decomposition numbers for weight three blocks of symmetric groups and Iwahori-Hecke algebras", Trans. Amer. Math. Soc. 360 (2008), no. 3, 1341–1376. 2. Fayers, "James's Conjecture holds for weight four blocks of Iwahori-Hecke algebras", J. Algebra 317 (2007), no. 2, 593–633. 3. Chuang and Tan, "Filtrations in Rouquier blocks of symmetric groups and Schur algebras", Proc. London Math. Soc. (3) 86 (2003), no. 3, 685–706. 4. James, Lyle and Mathas, "Rouquier blocks", Math. Z. 252 (2006), no. 3, 511–531. $\endgroup$
    – Andrew
    Commented May 8, 2021 at 9:56
  • $\begingroup$ @Andrew: Many thanks for these references. I can see that RoCK blocks will give a lower bound on $D_p(w)$, but why an upper bound? I know every block is derived equivalent to a RoCK block, but derived equivalences don't preserve decomposition numbers $\endgroup$
    – Mark Wildon
    Commented May 8, 2021 at 12:12
  • $\begingroup$ Sorry, I meant lower bound . Another approach would be to use the LLT polynomials, which again would give only a lower bound for the symmetric groups. You can also ask the same question for graded decomposition numbers, but much less is known about these although in principle they can be calculated using p-canonical bases. $\endgroup$
    – Andrew
    Commented May 8, 2021 at 22:09

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