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Finitely generated FI-modules have eventual character polynomials

Statement

Let $k$ be a field of characteristic $0$ and let $V$ be a finitely generated FI-module over $k$ with finite-dimensional values (The category FI and FI-modules over a commutative ring). Put $d : = weight (V)$ and $s : = stab-deg (V)$ ; both are finite because $V$ is finitely generated (Weight of an FI-module over a field of characteristic zero, Injectivity, surjectivity and stability degree). Then there is a unique character polynomial $P_{V} \in Q [X_{1}, X_{2}, \dots]$ of weighted degree at most $d$ such that

$χ_{V_{n}} (σ) = P_{V} (X_{1} (σ), X_{2} (σ), \dots)$

for every $n \geq s + d$ and every $σ \in S_{n}$ , where $χ_{V_{n}} (σ) : = tr (σ; V_{n})$ is the character of the $k S_{n}$ -module $V_{n}$ (Character polynomials in the cycle counts $X_{i}$ , The trace $tr (A)$ as the sum of the diagonal entries). No choice principle is used.

Facts & Assumptions

Given: A characteristic-zero field $k$ , a finitely generated FI-module $V$ over $k$ with finite-dimensional values, generated in degree $\leq g$ for some $g \geq 0$ , and the character functions $χ_{V_{n}} (σ) = tr (σ; V_{n})$ (Span, finite generation and generation degree of an FI-module, The trace $tr (A)$ as the sum of the diagonal entries).

[F1]

Generation in degree $\leq d$ is equivalent to the existence of a surjection $⨁_{i \in I} M (m_{i}) ↠ V$ with all $m_{i} \leq d$ , and $V$ is finitely generated exactly when some such presentation has finite index set (for some finite degree bound); $M (m) = M (k [S_{m}])$ is the free FI-module on the regular $k S_{m}$ -module (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).

[F2]

Generation in degree $\leq d$ implies $surj-deg (V) \leq d$ , and over a characteristic-zero field it implies $weight (V) \leq d$ (Bounded generation degree bounds surjectivity degree and weight).

[F3]

$inj-deg (M (W)) = 0$ for every FB-module $W$ (Free FI-modules have computable stability degree and weight).

[F4]

Every sub-FI-module of a finitely generated FI-module over a commutative Noetherian ring containing $Q$ is finitely generated (FI-modules over a Noetherian $Q$ -algebra form a Noetherian category).

[F5]

$Φ_{a}$ is exact over fields of characteristic $0$ , natural in the FI-module variable, and preserves finite direct sums, so the transition map of a direct sum is the direct sum of the transition maps (The coinvariant functors $Φ_{a}$ and the transition maps $T$ ).

[F6]

$stab-deg (V) = \max (inj-deg (V), surj-deg (V))$ , where $inj-deg$ and $surj-deg$ are the least thresholds beyond which all transition maps are injective, respectively surjective (Injectivity, surjectivity and stability degree).

[F7]

If an FI-module over a characteristic-zero field has finite-dimensional values and finite weight $d$ and stability degree $s$ , then its consistent sequence is uniformly representation stable with stable range $n \geq d + s$ ; consequently the multiplicity $c_{λ, n}$ of $V (λ)_{n}$ in $V_{n}$ is independent of $n$ for $n \geq d + s$ (Weight and stability degree give a uniform representation-stability range, Weight of an FI-module over a field of characteristic zero).

[F8]

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

[F9]

For a partition $λ$ with $∣ λ ∣ = m$ there is a character polynomial $P_{λ} \in Q [X_{1}, X_{2}, \dots]$ of weighted degree at most $m$ such that the character of the padded Specht module satisfies $χ^{λ [n]} (σ) = P_{λ} (X_{1} (σ), X_{2} (σ), \dots)$ for all $n \geq m + λ_{1}$ and all $σ \in S_{n}$ (Padded irreducible characters are eventually given by character polynomials, First-row padding of a partition).

[F10]

The field-valued Specht modules are the scalar extensions of the single $Z$ -model $S_{Z}^{λ}$ built from polytabloids (Integral and field-valued Specht modules), and the $k S_{n}$ -module $V (λ)_{n}$ is $k \otimes_{Q} S_{Q}^{λ [n]}$ , the irreducible with label $λ [n]$ (Weight of an FI-module over a field of characteristic zero, Specht modules classify the irreducibles over every field of characteristic zero). Consequently the trace of $σ \in S_{n}$ on $V (λ)_{n}$ is the image under $Z \to k$ of an integer and equals the value of the character polynomial of [F9] evaluated at the cycle counts of $σ$ .

[F11]

If two character polynomials agree as functions on $S_{n}$ for infinitely many $n$ , they are equal; the weighted degree of a sum is at most the maximum of the weighted degrees of the summands (Character polynomials in the cycle counts $X_{i}$ ).

[F12]

The trace on a finite direct sum is the sum of the block traces. Also $tr (A B) = \sum_{i, j} a_{i j} b_{j i} = tr (B A)$ , so $tr (B^{- 1} A B) = tr (A)$ for invertible $B$ . These follow from the diagonal-sum definition (The trace $tr (A)$ as the sum of the diagonal entries).

Proof

technique · direct

1.1F1F2F4

Since $V$ is generated in degree $\leq g$ , [F2] gives $weight (V) \leq g$ , so $d \leq g < \infty$ . By [F1] there is a finite surjection $M : = ⨁_{i = 1}^{r} M (m_{i}) ↠ V$ with all $m_{i} \leq g$ ; let $K$ be its kernel. The module $M$ is finitely generated, so $K$ is finitely generated by [F4] applied over the field $k$ ; say $K$ is generated in degree $\leq r^{'}$ .

2.1F2F3F5F6step 1.1

We show that $s = stab-deg (V) \leq \max (g, r^{'}) < \infty$ . For every $a$ the exactness of $Φ_{a}$ gives an exact sequence $0 \to Φ_{a} (K) \to Φ_{a} (M) \to Φ_{a} (V) \to 0$ by [F5]. The transition map of $M$ is the direct sum of the transition maps of the finitely many free modules $M (m_{i})$ , each injective in every degree by [F3] and [F5], so $T ⁣ : Φ_{a} (M)_{n} \to Φ_{a} (M)_{n + 1}$ is injective for all $n$ ; and $T ⁣ : Φ_{a} (K)_{n} \to Φ_{a} (K)_{n + 1}$ is surjective for $n \geq r^{'}$ by [F2]. Then $T ⁣ : Φ_{a} (V)_{n} \to Φ_{a} (V)_{n + 1}$ is injective for $n \geq r^{'}$ : a lift $\tilde{x}$ of $x$ with $T x = 0$ has $T \tilde{x}$ in the image of $Φ_{a} (K)$ , so $T \tilde{x} = T \tilde{y}$ for some $\tilde{y} \in Φ_{a} (K)_{n}$ , and injectivity on $Φ_{a} (M)$ gives $\tilde{x} = \tilde{y}$ , whence $x = 0$ . Hence $inj-deg (V) \leq r^{'}$ ; since also $surj-deg (V) \leq g$ by [F2], [F6] gives $s \leq \max (g, r^{'}) < \infty$ .

3.1F7F8step 2.1

By [F7] applied with the finite numbers $d$ and $s$ of step 2.1, the consistent sequence of $V$ is uniformly representation stable with stable range $n \geq s + d$ ; that is, for every $n \geq s + d$ the multiplicities $c_{λ, n}$ of the irreducibles $V (λ)_{n}$ in $V_{n}$ are independent of $n$ , so with $c_{λ} : = c_{λ, n}$ one has $V_{n} ≅ ⨁_{λ} c_{λ} V (λ)_{n}$ for all $n \geq s + d$ , the sum being finite because only labels with $∣ λ ∣ \leq d$ occur. Moreover every $λ$ with $c_{λ} \neq 0$ satisfies $∣ λ ∣ \leq d$ by the definition of weight and $λ_{1} \leq s$ by [F8]; hence $∣ λ ∣ + λ_{1} \leq d + s \leq n$ throughout the range.

4.1F9F10F11F12step 3.1

For each $λ$ with $c_{λ} \neq 0$ , [F9] and [F10] give a character polynomial $P_{λ}$ of weighted degree at most $∣ λ ∣$ with $χ_{V (λ)_{n}} (σ) = P_{λ} (X_{1} (σ), X_{2} (σ), \dots)$ for all $n \geq ∣ λ ∣ + λ_{1}$ and all $σ \in S_{n}$ . Define $P_{V} : = \sum_{λ} c_{λ} P_{λ}$ , a finite sum; every summand has weighted degree at most $d$ , so $P_{V}$ does too by [F11], including the empty sum $P_{V} = 0$ . For $n \geq s + d$ and $σ \in S_{n}$ , additivity of the trace over the direct sum [F12] gives $χ_{V_{n}} (σ) = \sum_{λ} c_{λ} χ_{V (λ)_{n}} (σ)$ , and by step 3.1 every summand satisfies $n \geq ∣ λ ∣ + λ_{1}$ , so this equals $\sum_{λ} c_{λ} P_{λ} (X (σ)) = P_{V} (X (σ))$ .

5.1F11step 4.1∎

The polynomial $P_{V}$ of step 4.1 has the required degree bound and represents $χ_{V_{n}}$ for all $n \geq s + d$ and all $σ \in S_{n}$ . It is unique: two character polynomials agreeing on $S_{n}$ for all $n \geq s + d$ agree on infinitely many $S_{n}$ , hence are equal by [F11]. This proves the theorem.

Depends on

Used by

Dimensions of finitely generated FI-modules are eventually polynomial Corollary

Dependency tree · two levels

82 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)
I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed., Oxford Mathematical Monographs, 1995 (author-hosted full text) (standard reference, not scraped)