nLab
length of an object

Contents

Contents

Idea

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

Definition

Let π’ž\mathcal{C} be an abelian category.

Definition

Given an object Xβˆˆπ’žX \in \mathcal{C}, then a Jordan-HΓΆlder sequence or composition series for XX is a finite filtration, i.e. a finite sequence of subobject inclusions into XX, starting with the zero objects

0=X 0β†ͺX 1β†ͺβ‹―β†ͺX nβˆ’1β†ͺX n=X 0 = X_0 \hookrightarrow X_1 \hookrightarrow \cdots \hookrightarrow X_{n-1} \hookrightarrow X_n = X

such that at each stage ii the quotient X i/X iβˆ’1X_i/X_{i-1} (i.e. the coimage of the monomorphism X iβˆ’1β†ͺX iX_{i-1} \hookrightarrow X_i) is a simple object of π’ž\mathcal{C}.

If a Jordan-HΓΆlder sequence for XX exists at all, then XX is said to be of finite length.

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

Proposition

(Jordan-HΓΆlder theorem)

If Xβˆˆπ’žX \in \mathcal{C} has finite length according to def. , then in fact all Jordan-HΓΆlder sequences for XX have the same length nβˆˆβ„•n \in \mathbb{N}.

(e.g. EGNO 15, theorem 1.5.4)

Definition

If an object Xβˆˆπ’žX \in \mathcal{C} has finite length according to def. , then the length nβˆˆβ„•n \in \mathbb{N} of any of its Jordan-HΓΆlder sequences, which is uniquely defined according to prop. , is called the length of the object XX.

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

Properties

Relation to Schur functors

In abelian categories that are also kk-linear tensor categories over a field kk 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 VV there exists an exterior power that vanishes, ∧ nV=0\wedge^n V = 0 (namely for all n>dim(V)n \gt dim(V)). Similarly, if VV is a super vector space of dimension (d,p)(d,p), then the combined (d+1)(d+1)st skew-symmetric tensor power and (p+1)(p+1)st symmetric tensor power annihilates it. In this way prop. below goes in the direction of establishing that in a kk-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 β€œtensor category”:

Definition

For kk an algebraically closed field of characteristic zero, then a kk-tensor category π’œ\mathcal{A} is an

  1. abelian

  2. rigid

  3. symmetric

  4. braided

  5. monoidal category

  6. enriched over kkMod = kkVect (i.e. kk-linear),

such that

  1. the tensor product functor βŠ—:π’œΓ—π’œβŸΆπ’œ\otimes \colon \mathcal{A} \times \mathcal{A} \longrightarrow \mathcal{A} is

    1. kModk Mod-enriched (i.e. kk-linear);

    2. exact

    in both arguments;

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

Such a kk-tensor category is called finitely βŠ—\otimes-generated if there exists an object Eβˆˆπ’œE \in \mathcal{A} such that every other object Xβˆˆπ’œX \in \mathcal{A} is a subquotient of a direct sum of tensor products E βŠ— nE^{\otimes^n}, for some nβˆˆβ„•n \in \mathbb{N}:

βŠ•iE βŠ— n i ↓ X β†ͺ (βŠ•iE βŠ— n i)/Q. \array{ && \underset{i}{\oplus} E^{\otimes^{n_i}} \\ && \downarrow \\ X &\hookrightarrow& (\underset{i}{\oplus} E^{\otimes^{n_i}})/Q } \,.

Such EE is called an βŠ—\otimes-generator for π’œ\mathcal{A}.

(Deligne 02, 0.1)

Definition

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

βˆ€Xβˆˆπ’œβˆƒNβˆˆβ„•βˆ€nβˆˆβ„•length(N βŠ— n)≀N 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 (π’œ,βŠ—)(\mathcal{A},\otimes) a kk-tensor category as in def., for Xβˆˆπ’œX \in \mathcal{A} an object, for nβˆˆβ„•n \in \mathbb{N} and Ξ»\lambda a partition of nn, say that the value of the Schur functor S Ξ»S_\lambda on XX is

S Ξ»(X)≔(V Ξ»βŠ—X βŠ— n) S n≔(1n!βˆ‘g∈S nρ(g))(V Ξ»βŠ—X βŠ— n) 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 nS_n is the symmetric group on nn elements;

  • V Ξ»V_\lambda is the irreducible representation of S nS_n corresponding to Ξ»\lambda;

  • ρ\rho is the diagional action of S nS_n on V Ξ»βŠ—X βŠ— nV_\lambda \otimes X^{\otimes_n}, coming from the canonical permutation action on X βŠ— nX^{\otimes_n};

  • (βˆ’) S n(-)^{S_n} denotes the subspace of invariants under the action ρ\rho

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

(Deligne 02, 1.4)

Proposition

For a tensor category π’œ\mathcal{A} the following are equivalent:

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

  2. For every object Xβˆˆπ’œX \in \mathcal{A} there exists nβˆˆβ„•n \in \mathbb{N} and a partition Ξ»\lambda of nn such that the corresponding value of the Schur functor, def. , on XX vanishes: S Ξ»(X)=0S_\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, CatΓ©gorie Tensorielle, Moscow Math. Journal 2 (2002) no. 2, 227-248. (pdf)

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

Last revised on March 9, 2018 at 13:38:23. See the history of this page for a list of all contributions to it.