Simultaneous core partitions

Definition\(s\)-cores of Young diagrams#✓fully proved

Let \(s\) be a natural number and \(\mu\) a Young diagram, with rows and columns numbered from \(0\); write \(r_i\) for the length of row \(i\) of \(\mu\) and \(c_j\) for the length of column \(j\) of \(\mu\). The hook length of the cell \((i, j)\) of \(\mu\) (row \(i\), column \(j\)) is \((r_i - j) + (c_j - i) - 1\): the cells to its right in row \(i\), the cells below it in column \(j\), and the cell itself (both differences count the cell, so one is subtracted). The diagram \(\mu\) is an \(s\)-core if no cell of \(\mu\) has hook length divisible by \(s\).

def IsCore (s : ℕ) (μ : YoungDiagram) : Prop :=
  ∀ c ∈ μ, ¬ s ∣ (μ.rowLen c.1 - c.2) + (μ.colLen c.2 - c.1) - 1

Used by 1

Used byContainment of simultaneous cores (Olsson–Stanton question)

DefinitionContainment of simultaneous cores (Olsson–Stanton question)#✓fully proved

The proposition that for all coprime positive integers \(s\) and \(t\) and every Young diagram \(\kappa\) that is both an \(s\)-core and a \(t\)-core and has at least as many cells as every Young diagram that is both an \(s\)-core and a \(t\)-core, every Young diagram that is both an \(s\)-core and a \(t\)-core is contained in \(\kappa\), that is, each of its cells is a cell of \(\kappa\).

The proposition does not assert that such a \(\kappa\) exists. In the report, \(\kappa_{s,t}\) is the partition of maximum size that is both an \(s\)-core and a \(t\)-core, and the question is whether the Young diagram of every such partition is contained in that of \(\kappa_{s,t}\).

def simultaneousCoreContainment : Prop :=
  ∀ s t : ℕ, 0 < s → 0 < t → Nat.Coprime s t →
    ∀ κ : YoungDiagram, IsCore s κ → IsCore t κ →
      (∀ ν : YoungDiagram, IsCore s ν → IsCore t ν → ν.card ≤ κ.card) →
        ∀ μ : YoungDiagram, IsCore s μ → IsCore t μ → μ ≤ κ

Uses 1 · Used by 1

Statement uses\(s\)-cores of Young diagrams

Used byFayers–Vandehey containment theorem for simultaneous cores

TheoremFayers–Vandehey containment theorem for simultaneous cores#●ready to prove

Call a Young diagram an \(s\)-core when none of its cells has hook length divisible by \(s\). For all coprime positive integers \(s\) and \(t\), if \(\kappa\) is a Young diagram that is both an \(s\)-core and a \(t\)-core and has at least as many cells as every Young diagram that is both an \(s\)-core and a \(t\)-core, then every Young diagram that is both an \(s\)-core and a \(t\)-core is contained in \(\kappa\) (each of its cells is a cell of \(\kappa\)).

Fayers (Theorem 5.1, first proved by Vandehey) shows that for coprime positive integers \(s\) and \(t\) every simultaneous \(s\)- and \(t\)-core is contained in \(\kappa_{s,t}\), the simultaneous core of the largest size \((s^2 - 1)(t^2 - 1)/24\), answering the question of Olsson and Stanton. A simultaneous core \(\kappa\) of maximum size is then contained in \(\kappa_{s,t}\) and has at least as many cells, so it equals \(\kappa_{s,t}\) and contains every simultaneous core.

public theorem_wanted simultaneousCoreContainment_holds : simultaneousCoreContainment

Sources

• J. B. Olsson (joint with D. Stanton, C. Bessenrodt, G. Navarro, P. H. Tiep), On block identities and block inclusions, in Representations of Finite Groups, Oberwolfach Reports 3 (2006), no. 15, p. 912 ff., section 2 (Block identities), DOI 10.4171/OWR/2006/15; resolution: M. Fayers, The t-core of an s-core, J. Combin. Theory Ser. A 118 (2011), 1525-1539, DOI 10.1016/j.jcta.2011.01.004, arXiv:1007.1071, Section 5, Theorem 5.1 ("Suppose s and t are coprime positive integers, and λ is an (s, t)-core. Then [λ] ⊆ [κ_{s,t}]"), proved in Proposition 5.5; Fayers cites the first proof as J. Vandehey, Containment in (s, t)-core partitions, arXiv:0809.2134, Theorem 2.4, https://arxiv.org/abs/1007.1071.

Uses 1

Statement usesContainment of simultaneous cores (Olsson–Stanton question)

2 fully proved1 ready to prove

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