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Weight and stability degree give a uniform representation-stability range

Statement

Let $k$ be a field of characteristic $0$ and let $V$ be an FI-module over $k$ with finite-dimensional values, weight $d : = weight (V) < \infty$ and stability degree $s : = stab-deg (V) < \infty$ (Weight of an FI-module over a field of characteristic zero, Injectivity, surjectivity and stability degree). Let $(V_{n}, φ_{n})$ be the consistent sequence determined by $V$ (Uniform representation stability for consistent sequences of $S_{n}$ -modules). Then $(V_{n}, φ_{n})$ is uniformly representation stable with stable range $n \geq d + s$ : the maps $φ_{n}$ are injective for all $n \geq d + s$ , the $S_{n + 1}$ -span of $φ_{n} (V_{n})$ is all of $V_{n + 1}$ for all $n \geq d + s$ , and for every partition $λ$ the multiplicity $c_{λ, n}$ of $V (λ)_{n}$ in $V_{n}$ is independent of $n$ for all $n \geq d + s$ . No choice principle is used.

Facts & Assumptions

Given: A characteristic-zero field $k$ , an FI-module $V$ over $k$ with finite-dimensional values, $d = weight (V) < \infty$ and $s = stab-deg (V) < \infty$ , and the consistent sequence $(V_{n}, φ_{n})$ with $φ_{n}$ induced by the standard inclusion $[n] ↪ [n + 1]$ (Uniform representation stability for consistent sequences of $S_{n}$ -modules); for $n \geq ∣ λ ∣ + λ_{1}$ , $V (λ)_{n}$ is the irreducible $k S_{n}$ -module with label $λ [n]$ , of multiplicity $c_{λ, n}$ in $V_{n}$ (Weight of an FI-module over a field of characteristic zero, Specht modules classify the irreducibles over every field of characteristic zero).

[F1]

A consistent sequence is uniformly representation stable with stable range $n \geq N$ when (I) $φ_{n}$ is injective for all $n \geq N$ , (II) the $S_{n + 1}$ -span of $φ_{n} (V_{n})$ is $V_{n + 1}$ for all $n \geq N$ , and (III) for every partition $λ$ the multiplicity $c_{λ, n}$ is independent of $n$ for all $n \geq N$ (Uniform representation stability for consistent sequences of $S_{n}$ -modules).

[F2]

$stab-deg (V) = \max (inj-deg (V), surj-deg (V))$ , so for every $a \geq 0$ the transition map $T ⁣ : Φ_{a} (V)_{n} \to Φ_{a} (V)_{n + 1}$ is an isomorphism for all $n \geq s$ (Injectivity, surjectivity and stability degree, The coinvariant functors $Φ_{a}$ and the transition maps $T$ ).

[F3]

$weight (V) = d$ means that no irreducible of label $λ$ with $∣ λ ∣ > d$ occurs as a constituent of $V_{n}$ for any $n \geq ∣ λ ∣ + λ_{1}$ ; hence $c_{λ, n} = 0$ whenever $∣ λ ∣ > d$ and $c_{λ, n}$ is defined (Weight of an FI-module over a field of characteristic zero).

[F4]

For $μ = λ [n] ⊢ n$ and $a \leq n$ : (i) $(V_{μ})_{S_{n - a}} = 0 ⟺ a < ∣ λ ∣$ ; (ii) if $a = ∣ λ ∣$ , then $(V_{μ})_{S_{n - a}} ≅ V_{λ}$ ; (iii) for fixed $λ$ and $a \geq ∣ λ ∣$ , $(V (λ)_{n})_{S_{n - a}} ≅ {Ind}_{S_{∣ λ ∣} \times S_{a - ∣ λ ∣}}^{S_{a}} (V_{λ} ⊠ k)$ for all $n \geq a + λ_{1}$ ; (iv) if an FI-module $U$ over a characteristic-zero field has $weight (U) \leq a$ , then, for every $n \geq a$ , every subquotient $W_{n}$ of $U_{n}$ satisfies $W_{n} = 0 ⟺ (W_{n})_{S_{n - a}} = 0$ (Coinvariants detect vanishing of subquotients of bounded weight, and padded irreducibles stabilise).

[F5]

Every constituent $V (λ)_{n}$ of $V_{n}$ satisfies $λ_{1} \leq s$ (A stability degree bounds the first row of every padding tail).

[F6]

$Φ_{a} (U)_{n} = (U_{a ⊔ n})_{S_{n}}$ with transition map $T$ induced by an injection $a ⊔ n \to a ⊔ n + 1$ fixing $a$ ; $Φ_{a}$ is a functor, it is exact when $k$ contains $Q$ , and it preserves finite direct sums, so $(U \oplus U^{'})_{S_{n}} ≅ U_{S_{n}} \oplus U_{S_{n}}^{'}$ (The coinvariant functors $Φ_{a}$ and the transition maps $T$ , The coinvariants functor).

[F7]

Over the characteristic-zero field $k$ , every finite-dimensional $k S_{n}$ -module is a direct sum of copies of the pairwise non-isomorphic irreducibles $V (λ)_{n}$ (Maschke's theorem for finite groups over fields whose characteristic does not divide $∣ G ∣$ , Specht modules classify the irreducibles over every field of characteristic zero).

Proof

technique · direct

1.1F1F6given

The commuting $S_{a}$ -action on $Φ_{a} (V)$ makes every $T$ $S_{a}$ -equivariant by The coinvariant functors $Φ_{a}$ and the transition maps $T$ , so the isomorphisms used below preserve representation types. Let $K_{n} : = \ker φ_{n} \subseteq V_{n}$ and let $φ_{n}^{'} : = {Ind}_{S_{n}}^{S_{n + 1}} φ_{n} ⁣ : {Ind}_{S_{n}}^{S_{n + 1}} V_{n} \to V_{n + 1}$ with cokernel $C_{n + 1}$ , so that $C_{n + 1}$ is a quotient of $V_{n + 1}$ . The image of $φ_{n}^{'}$ is the $S_{n + 1}$ -span of $φ_{n} (V_{n})$ , since $φ_{n}^{'} (σ \otimes v) = σ \cdot φ_{n} (v)$ ; hence condition (I) is $K_{n} = 0$ for all $n \geq N$ and condition (II) is $C_{n + 1} = 0$ for all $n \geq N$ .

2.1F3step 1.1

If $∣ λ ∣ > d$ , then $c_{λ, n} = 0$ for every $n$ for which $c_{λ, n}$ is defined, so condition (III) is automatic for such $λ$ ; it remains to treat $∣ λ ∣ \leq d$ .

2.2F3F4step 1.1

Vanishing criteria: for $n \geq d$ one has $K_{n} = 0 ⟺ (K_{n})_{S_{n - d}} = 0$ , and for $n + 1 \geq d$ one has $C_{n + 1} = 0 ⟺ (C_{n + 1})_{S_{n + 1 - d}} = 0$ . Indeed $K_{n}$ is a submodule of $V_{n}$ and $C_{n + 1}$ is a quotient of $V_{n + 1}$ , both are subquotients of modules of weight $d$ (Weight of an FI-module over a field of characteristic zero), and F4 applies with $a = d$ .

2.3F6givenstep 1.1

Factorization: for $n \geq d$ , identify $Φ_{d} (V)_{n - d}$ with $(V_{n})_{S_{n - d}}$ and $Φ_{d} (V)_{n + 1 - d}$ with $(V_{n + 1})_{S_{n + 1 - d}}$ along the standard identifications of $d ⊔ (n - d)$ with $[n]$ and of $d ⊔ (n + 1 - d)$ with $[n + 1]$ . The transition map $T ⁣ : (V_{n})_{S_{n - d}} \to (V_{n + 1})_{S_{n + 1 - d}}$ factors in two ways: $T = q \circ (φ_{n})_{S_{n - d}}$ , where $q$ is the quotient map onto coinvariants of the larger subgroup $S_{n + 1 - d} \geq S_{n - d}$ , and $T = (φ_{n}^{'})_{S_{n + 1 - d}} \circ ψ$ , where $ψ$ is induced by $v \mapsto 1 \otimes v$ . Exactness of coinvariants over $k$ gives $(K_{n})_{S_{n - d}} = \ker ((φ_{n})_{S_{n - d}})$ and $(C_{n + 1})_{S_{n + 1 - d}} = coker ((φ_{n}^{'})_{S_{n + 1 - d}})$ .

3.1F2F4F5F6F7step 2.1

We prove (III) by induction on $m = ∣ λ ∣$ in the range $m \leq d$ . Fix $m \leq d$ , let $n \geq s + d$ (so $n - m \geq s \geq 0$ ), and assume the multiplicities $c_{μ, n^{'}}$ are independent of $n^{'}$ in the range for all $∣ μ ∣ < m$ . Since $n - m \geq s$ , the transition maps give isomorphisms $Φ_{m} (V)_{n - m} ≅ Φ_{m} (V)_{i}$ for all $i \geq s$ , so the isomorphism class of $(V_{n})_{S_{n - m}} = Φ_{m} (V)_{n - m}$ is independent of $n$ . By [F7] $V_{n} = ⨁_{λ} c_{λ, n} V (λ)_{n}$ is a finite direct sum of irreducibles, and [F6] gives $(V_{n})_{S_{n - m}} = ⨁_{λ} c_{λ, n} (V (λ)_{n})_{S_{n - m}}$ ; by F4 with $a = m$ only the labels with $∣ λ ∣ \leq m$ contribute, and by F4 with $a = m$ those with $∣ λ ∣ = m$ contribute $(V (λ)_{n})_{S_{n - m}} ≅ V_{λ}$ . For $∣ λ ∣ < m$ , F4 gives $(V (λ)_{n})_{S_{n - m}} ≅ {Ind}_{S_{∣ λ ∣} \times S_{m - ∣ λ ∣}}^{S_{m}} (V_{λ} ⊠ k)$ , independent of $n$ , provided $n \geq m + λ_{1}$ ; this holds because $λ_{1} \leq s$ for every occurring constituent by [F5] and $m + λ_{1} \leq d + s \leq n$ . Hence in the decomposition $(V_{n})_{S_{n - m}} = ⨁_{∣ λ ∣ < m} c_{λ, n} (V (λ)_{n})_{S_{n - m}} \oplus ⨁_{∣ λ ∣ = m} c_{λ, n} V_{λ}$ the first summand has an isomorphism class independent of $n$ (each factor is independent of $n$ and each multiplicity is constant by the induction hypothesis), and the total has an isomorphism class independent of $n$ , so the second summand does too; as the $V_{λ}$ with $∣ λ ∣ = m$ are pairwise non-isomorphic, every multiplicity $c_{λ, n}$ with $∣ λ ∣ = m$ is independent of $n$ in the range. This proves (III) for all $λ$ with $∣ λ ∣ \leq d$ and hence, with step 2.1, for all $λ$ .

3.2F2step 1.1step 2.2step 2.3

Let $n \geq s + d$ , so that $n - d \geq s$ and $T ⁣ : Φ_{d} (V)_{n - d} \to Φ_{d} (V)_{n + 1 - d}$ is an isomorphism by [F2]. The first factorization in step 2.3 then forces $(φ_{n})_{S_{n - d}}$ to be injective, and the second forces $(φ_{n}^{'})_{S_{n + 1 - d}}$ to be surjective; by the exactness identifications in step 2.3 and the criteria in step 2.2, $K_{n} = 0$ and $C_{n + 1} = 0$ . By step 1.1, conditions (I) and (II) hold for all $n \geq s + d$ .

4.1step 2.1step 3.1step 2.2step 2.3step 3.2∎

Steps 2.1 and 3.1 give condition (III) for $n \geq d + s$ and steps 2.2, 2.3 and 3.2 give conditions (I) and (II) for $n \geq d + s$ ; hence the consistent sequence $(V_{n}, φ_{n})$ is uniformly representation stable with stable range $n \geq d + s$ .

Depends on

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38 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Thomas Church, Jordan S. Ellenberg and Benson Farb, “FI-modules and stability for representations of symmetric groups”, arXiv:1204.4533v4 (2014); published in Duke Math. J. 164 (2015) 1833–1910 (standard reference, not scraped)
Thomas Church, Jordan S. Ellenberg, Benson Farb and Rohit Nagpal, “FI-modules over Noetherian rings”, Geom. Topol. 18 (2014) 2951–2984; author-hosted full text (standard reference, not scraped)