nLab length of an object

Idea

The concept of length of an object in an abelian category $\mathcal{C}$ is akin to the concept of dimension of vector spaces, to which it reduces in the case that $\mathcal{C} =$ Vect. The 1-dimensional vector space is a simple object in Vect, and the dimension of a vector space $V$, if it is finite, may be thought of as the number of times that one may split off such a simple object from $V$. The definition of length generalizes this concept, notably to modules over some ring.

Definition

Let $\mathcal{C}$ be an abelian category.

Definition

Given an object $X \in \mathcal{C}$, then a Jordan-Hölder sequence or composition series for $X$ is a finite filtration, i.e. a finite sequence of subobject inclusions into $X$, starting with the zero objects

$$0 = X_0 \hookrightarrow X_1 \hookrightarrow \cdots \hookrightarrow X_{n-1} \hookrightarrow X_n = X$$

such that at each stage $i$ the quotient $X_i/X_{i-1}$ (i.e. the cokernel of the monomorphism $X_{i-1} \hookrightarrow X_i$) is a simple object of $\mathcal{C}$.

If a Jordan-Hölder sequence for $X$ exists at all, then $X$ is said to be of finite length.

(e.g. EGNO 15, def. 1.5.3)

Proposition

(Jordan-Hölder theorem)

If $X \in \mathcal{C}$ has finite length according to def. , then in fact all Jordan-Hölder sequences for $X$ have the same length $n \in \mathbb{N}$.

(e.g. EGNO 15, theorem 1.5.4)

Definition

If an object $X \in \mathcal{C}$ has finite length according to def. , then the length $n \in \mathbb{N}$ of any of its Jordan-Hölder sequences, which is uniquely defined according to prop. , is called the length of the object $X$.

(e.g. EGNO 15, def. 1.5.5)

Properties

Relation to Schur functors

In abelian categories that are also $k$-linear tensor categories over a field $k$ of characteristic zero, then objects have finite length precisely if they are annihilated by some Schur functor for the symmetric group.

This is a (considerable) generalization of the familiar fact that for every finite dimensional vector space $V$ there exists an exterior power that vanishes, $\wedge^n V = 0$ (namely for all $n \gt dim(V)$). Similarly, if $V$ is a super vector space of dimension $(d,p)$, then the combined $(d+1)$st skew-symmetric tensor power and $(p+1)$st symmetric tensor power annihilates it. In this way prop. below goes in the direction of establishing that in a $k$-linear tensor category all objects of bounded length , in the sense of def. , behave like having underlying super vector spaces. The completion of this statement is Deligne's theorem on tensor categories, see there for more.

First we need to fix the precise meaning of “tensor category”:

Definition

For $k$ an algebraically closed field of characteristic zero, then a $k$-tensor category $\mathcal{A}$ is an

1. abelian

2. rigid

3. symmetric

4. braided

5. monoidal category

6. enriched over $k$Mod = $k$Vect (i.e. $k$-linear),

such that

1. the tensor product functor $\otimes \colon \mathcal{A} \times \mathcal{A} \longrightarrow \mathcal{A}$ is

   1. $k Mod$-enriched (i.e. $k$-linear);

   2. exact

   in both arguments;

2. $End(1) \simeq k$ (the endomorphism ring of the tensor unit coincides with $k$).

Such a $k$-tensor category is called finitely $\otimes$-generated if there exists an object $E \in \mathcal{A}$ such that every other object $X \in \mathcal{A}$ is a subquotient of a direct sum of tensor products $E^{\otimes^n}$, for some $n \in \mathbb{N}$:

$$\array{ && \underset{i}{\oplus} E^{\otimes^{n_i}} \\ && \downarrow \\ X &\hookrightarrow& (\underset{i}{\oplus} E^{\otimes^{n_i}})/Q } \,.$$

Such $E$ is called an $\otimes$-generator for $\mathcal{A}$.

(Deligne 02, 0.1)

Definition

A tensor category $\mathcal{A}$ (def. ) is said to have subexponential growth if for every object $X$ there exists a natural number $N$ such that $X$ is of length (def. ) at most $N$, and that also all tensor product powers of $X$ are of length bounded by the corresponding powers of $N$:

$$\underset{X \in \mathcal{A}}{\forall} \underset{N \in \mathbb{N}}{\exists} \underset{n \in \mathbb{N}}{\forall} \; length(N^{\otimes^n}) \leq N^n \,.$$

(e.g. EGNO 15, def. 9.11.1)

Definition

For $(\mathcal{A},\otimes)$ a $k$-tensor category as in def., for $X \in \mathcal{A}$ an object, for $n \in \mathbb{N}$ and $\lambda$ a partition of $n$, say that the value of the Schur functor $S_\lambda$ on $X$ is

$$S_{\lambda}(X) \coloneqq (V_\lambda \otimes X^{\otimes_n})^{S_n} \coloneqq \left( \frac{1}{n!} \underset{g\in S_n}{\sum} \rho(g) \right) \left( V_\lambda \otimes X^{\otimes_n} \right)$$

where

• $S_n$ is the symmetric group on $n$ elements;

• $V_\lambda$ is the irreducible representation of $S_n$ corresponding to $\lambda$;

• $\rho$ is the diagional action of $S_n$ on $V_\lambda \otimes X^{\otimes_n}$, coming from the canonical permutation action on $X^{\otimes_n}$;

• $(-)^{S_n}$ denotes the subspace of invariants under the action $\rho$

• the last expression just rewrites this as a group averaging.

(Deligne 02, 1.4)

Proposition

For a tensor category $\mathcal{A}$ the following are equivalent:

1. the category has subexponential growth (def. ).

2. For every object $X \in \mathcal{A}$ there exists $n \in \mathbb{N}$ and a partition $\lambda$ of $n$ such that the corresponding value of the Schur functor, def. , on $X$ vanishes: $S_\lambda(X) = 0$.

(Deligne 02, prop. 05)

References

• Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik, section 1.5 in Tensor categories, Mathematical Surveys and Monographs, Volume 205, American Mathematical Society, 2015 (pdf

  ))

• Wikipedia, Composition series

The relation to Schur functors is discussed in

• Pierre Deligne, Catégorie Tensorielle, Moscow Math. Journal 2 (2002) no. 2, 227-248. (pdf)

For more on this see at Deligne's theorem on tensor categories.