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A stability degree bounds the first row of every padding tail

Statement

Read missing partition parts as $0$ when comparing row lengths.

Let $k$ be a field of characteristic $0$ , let $V$ be an FI-module over $k$ (The category FI and FI-modules over a commutative ring), and put $s : = stab-deg (V) < \infty$ (Injectivity, surjectivity and stability degree). Then for every $n \geq 0$ , every irreducible constituent $V (λ)_{n}$ of the $k S_{n}$ -module $V_{n}$ satisfies

$λ_{1} \leq s .$

Equivalently, since the padded partition is $λ [n] = (n - ∣ λ ∣, λ_{1}, λ_{2}, \dots)$ , the second row of the Specht diagram $λ [n]$ has length at most $s$ ; the first row of the padding tail is the second row of the full diagram (First-row padding of a partition, Weight of an FI-module over a field of characteristic zero). No choice principle is used.

Facts & Assumptions

Given: A field $k$ of characteristic $0$ , an FI-module $V$ over $k$ with $s = stab-deg (V) < \infty$ , and for $m \geq 0$ the sub-FI-module $F^{m} V : = {span}_{V} (V_{\leq m})$ , with $F^{- 1} V : = 0$ , the graded quotient $G_{m} : = F^{m} V / F^{m - 1} V$ and its degree- $m$ part $H_{m} : = (G_{m})_{m}$ ; for a partition $μ ⊢ m$ , $V_{μ}$ is the irreducible $k S_{m}$ -module with label $μ$ (Specht modules classify the irreducibles over every field of characteristic zero, Weight of an FI-module over a field of characteristic zero, Quotient module $M / N$ with scalar multiplication on additive cosets).

[F1]

${span}_{V} (V_{\leq m})$ is the smallest sub-FI-module containing all values of $V$ in degrees $\leq m$ ; it is generated in degree $\leq m$ , the inclusions $F^{m} V \subseteq F^{m + 1} V$ hold, and $V = ⋃_{m \geq 0} F^{m} V$ because every element of $V_{S}$ is the bijection transport of an element of $V_{[∣ S ∣]}$ and hence lies in $(F^{∣ S ∣} V)_{S}$ (Span, finite generation and generation degree of an FI-module).

[F2]

Every quotient $U ↠ W$ of FI-modules satisfies $surj-deg (W) \leq surj-deg (U)$ (Bounded generation degree bounds surjectivity degree and weight); generation in degree $\leq d$ passes to quotients, and for such a quotient the evaluation map $⨁_{i \leq d} M (W_{i}) ↠ W$ is surjective (Generation in bounded degree presents an FI-module as a quotient of free FI-modules, The free FI-module $M (W)$ generated by an $S_{m}$ -module).

[F3]

For $μ ⊢ m$ and $0 \leq a \leq m$ , the coinvariants $(V_{μ})_{S_{m - a}}$ vanish if and only if $m - a > μ_{1}$ , i.e. $μ_{1} < m - a$ (Coinvariants detect vanishing of subquotients of bounded weight, and padded irreducibles stabilise, The coinvariant functors $Φ_{a}$ and the transition maps $T$ ).

[F4]

For a partition $μ ⊢ m$ and $n \geq m$ , the constituents of $M (V_{μ})_{n} ≅ {Ind}_{S_{m} \times S_{n - m}}^{S_{n}} (V_{μ} ⊠ k)$ are the $V_{ν}$ with $ν ⊢ n$ obtained from $μ$ by adding one box in each of $n - m$ different columns, so $ν_{1} \geq ν_{2}$ and $ν_{2} \leq μ_{1}$ ; and if $ν = λ [n]$ then $ν_{2} = λ_{1}$ (Branching rule for Specht modules: adding or removing boxes in distinct columns, The free FI-module $M (W)$ generated by an $S_{m}$ -module, First-row padding of a partition).

[F5]

$Φ_{a} (V)_{n} = (V_{a ⊔ n})_{S_{n}}$ , and $T ⁣ : Φ_{a} (V)_{n} \to Φ_{a} (V)_{n + 1}$ is the map induced by an injection $a ⊔ n \to a ⊔ n + 1$ fixing $a$ ; $stab-deg (V) = \max (inj-deg (V), surj-deg (V))$ , so $surj-deg (V) \leq s$ (The coinvariant functors $Φ_{a}$ and the transition maps $T$ , Injectivity, surjectivity and stability degree).

[F6]

Over the characteristic-zero field $k$ , every finite-dimensional $k S_{m}$ -module is a direct sum of irreducibles $V_{μ}$ (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), and $M$ is additive in its argument, so $M (⨁_{i} N_{i}) ≅ ⨁_{i} M (N_{i})$ (The free FI-module $M (W)$ generated by an $S_{m}$ -module).

Proof

technique · direct

1.1F1given

The $F^{m} V$ are sub-FI-modules with $F^{m} V \subseteq F^{m + 1} V$ , $V = ⋃_{m} F^{m} V$ , and each $F^{m} V$ is generated in degree $\leq m$ .

1.2F1F6givenalgebra

If every constituent of every $(G_{m})_{n}$ has padding tail first row $\leq s$ , then so does every constituent of every $V_{n}$ : let $V (λ)_{n} ≅ P / Q$ be a constituent of $V_{n}$ and lift a $k$ -basis of the finite-dimensional module $P / Q$ to elements of $P$ and take the span of their finite $S_{n}$ -orbits, giving a finite-dimensional $k S_{n}$ -submodule $P_{0} \subseteq P$ with $P_{0} + Q = P$ ; then $P_{0} / (P_{0} \cap Q) ≅ P / Q$ , and $P_{0} \subseteq (F^{M} V)_{n}$ for some $M$ because a finite basis of $P_{0}$ lies in a common filtration term. Intersecting the filtration with $P_{0}$ gives finite-dimensional successive factors embedding in $(G_{m})_{n}$ . Maschke splits this finite filtration, so every simple constituent of $P_{0}$ occurs in one of those factors; hence $V (λ)_{n}$ is a constituent of some $(G_{m})_{n}$ .

1.3given

Suppose $m \leq s$ . Then every constituent $V_{μ}$ of $H_{m}$ has $μ_{1} \leq m \leq s$ because $μ ⊢ m$ .

1.4F2F3F5given

Suppose $m \geq s + 1$ and set $\overline{V} : = V / F^{m - 1} V$ and $a_{0} : = m - s - 1 \geq 0$ . As a quotient of $V$ , the FI-module $\overline{V}$ has $surj-deg (\overline{V}) \leq s$ by [F2], so $T ⁣ : Φ_{a_{0}} (\overline{V})_{s} \to Φ_{a_{0}} (\overline{V})_{s + 1}$ is surjective. Its domain is $({\overline{V}}_{m - 1})_{S_{s}} = 0$ because ${\overline{V}}_{m - 1} = 0$ , and its target is $({\overline{V}}_{m})_{S_{s + 1}}$ ; also ${\overline{V}}_{m} = V_{m} / {span}_{V} (V_{< m})_{m} ≅ H_{m}$ by the second isomorphism theorem. Hence $(H_{m})_{S_{s + 1}} = 0$ , and for every constituent $V_{μ}$ of $H_{m}$ the coinvariants $(V_{μ})_{S_{s + 1}}$ are a subquotient of $(H_{m})_{S_{s + 1}}$ , hence vanish; by [F3] applied with $a = m - (s + 1)$ , so that $m - a = s + 1$ , this says $s + 1 > μ_{1}$ , i.e. $μ_{1} \leq s$ .

2.1F1F2step 1.1

The quotient $G_{m}$ is generated in degree $\leq m$ and has $(G_{m})_{i} = 0$ for $i < m$ , so by [F2] the evaluation map $M (H_{m}) ↠ G_{m}$ is surjective. Hence every constituent of $(G_{m})_{n}$ is a constituent of $M (H_{m})_{n}$ .

2.2F1F4F6step 1.4

Every constituent of $M (H_{m})_{n}$ has padding tail first row $\leq s$ : let $V (λ)_{n} ≅ P / Q$ be a constituent of $M (H_{m})_{n}$ and choose $x \in P ∖ Q$ ; the $S_{n}$ -orbit span $X$ of $x$ is finite-dimensional, and $X / (X \cap Q)$ is a nonzero submodule of the simple $V (λ)_{n}$ , hence isomorphic to it, so $V (λ)_{n}$ is a constituent of $X$ . Since $X$ lies in the direct sum $⨁_{T \subseteq [n], ∣ T ∣ = m} H_{m} (T)$ over finitely many $T$ , after transporting by bijections $[m] \to T$ for these finitely many subsets, the projections of $X$ to the summands $H_{m} (T)$ generate a finite-dimensional $k S_{m}$ -submodule $N \subseteq H_{m}$ with $X \subseteq M (N)_{n}$ ; so $V (λ)_{n}$ is a constituent of $M (N)_{n}$ . By [F6] write $N ≅ ⨁_{μ} V_{μ}^{\oplus c_{μ}}$ (finite sum over partitions $μ ⊢ m$ with $c_{μ} \neq 0$ ); every such $V_{μ}$ is a constituent of $H_{m}$ , so $μ_{1} \leq s$ by steps 1.3 and 1.4. By [F6] $M (N) ≅ ⨁_{μ} M (V_{μ})^{\oplus c_{μ}}$ , and by [F4] every constituent $V_{ν}$ of $M (V_{μ})_{n}$ satisfies $ν_{2} \leq μ_{1} \leq s$ ; a constituent of a finite direct sum is a constituent of one summand, so for the constituent $V (λ)_{n} ≅ V_{λ [n]}$ of $M (N)_{n}$ we get $λ_{1} = (λ [n])_{2} \leq s$ .

3.1step 1.2step 1.3step 1.4step 2.1step 2.2∎

Steps 2.1 and 2.2 show that every constituent of every $(G_{m})_{n}$ has padding tail first row at most $s$ (for $m \leq s$ via step 1.3, for $m \geq s + 1$ via step 1.4), and step 1.2 transfers this bound to every constituent $V (λ)_{n}$ of $V_{n}$ ; thus $λ_{1} \leq s$ always. Since $λ [n] = (n - ∣ λ ∣, λ_{1}, λ_{2}, \dots)$ , the number $λ_{1}$ is the length of the second row of the Specht diagram $λ [n]$ . No choice was used.

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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)
I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed., Oxford Mathematical Monographs, 1995 (author-hosted full text) (standard reference, not scraped)