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\begin{document}

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\section*{length of an object}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{additive_and_abelian_categories}{}\paragraph*{{Additive and abelian categories}}\label{additive_and_abelian_categories}

[[!include additive and abelian categories - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{RelationToSchurFunctors}{Relation to Schur functors}\dotfill \pageref*{RelationToSchurFunctors} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The concept of \emph{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 \emph{length} generalizes this concept, notably to [[modules]] over some [[ring]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

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

\begin{defn}
\label{FiniteLength}\hypertarget{FiniteLength}{}
Given an [[object]] $X \in \mathcal{C}$, then a \emph{Jordan-H\"o{}lder sequence} or \emph{[[composition series]]} for $X$ is a finite [[filtration]], i.e. a finite sequence of [[subobject]] inclusions into $X$, starting with the [[zero objects]]

\begin{displaymath}
0 = X_0 \hookrightarrow X_1 \hookrightarrow \cdots \hookrightarrow X_{n-1} \hookrightarrow X_n = X
\end{displaymath}
such that at each stage $i$ the [[quotient]] $X_i/X_{i-1}$ (i.e. the [[coimage]] of the [[monomorphism]] $X_{i-1} \hookrightarrow X_i$) is a [[simple object]] of $\mathcal{C}$.

If a Jordan-H\"o{}lder sequence for $X$ exists at all, then $X$ is said to be of \emph{finite length}.

\end{defn}
(e.g. \hyperlink{EGNO15}{EGNO 15, def. 1.5.3})

\begin{prop}
\label{JordanHolderSequenceHasDefiniteLength}\hypertarget{JordanHolderSequenceHasDefiniteLength}{}
\textbf{([[Jordan-Hölder theorem]])}

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

\end{prop}
(e.g. \hyperlink{EGNO15}{EGNO 15, theorem 1.5.4})

\begin{defn}
\label{LengthOfAnObject}\hypertarget{LengthOfAnObject}{}
If an object $X \in \mathcal{C}$ has finite length according to def. \ref{FiniteLength}, then the length $n \in \mathbb{N}$ of any of its Jordan-H\"o{}lder sequences, which is uniquely defined according to prop. \ref{JordanHolderSequenceHasDefiniteLength}, is called the \emph{length of the object} $X$.

\end{defn}
(e.g. \hyperlink{EGNO15}{EGNO 15, def. 1.5.5})

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{RelationToSchurFunctors}{}\subsubsection*{{Relation to Schur functors}}\label{RelationToSchurFunctors}

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 [[symmetric algebra|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. \ref{LenghtOfObjectIsBounded} below goes in the direction of establishing that in a $k$-linear tensor category all objects of bounded length , in the sense of def. \ref{LenghtOfObjectIsBounded}, 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]]'':

\begin{defn}
\label{TensorCategory}\hypertarget{TensorCategory}{}
For $k$ an [[algebraically closed field]] of [[characteristic zero]], then a \emph{$k$-[[tensor category]]} $\mathcal{A}$ is an

\begin{enumerate}%
\item [[abelian category|abelian]]


\item [[rigid monoidal category|rigid]]


\item [[symmetric monoidal category|symmetric]]


\item [[braided monoidal category|braided]]


\item [[monoidal category]]


\item [[enriched category|enriched]] over $k$[[Mod]] = $k$[[Vect]] (i.e. $k$-linear),



\end{enumerate}
such that

\begin{enumerate}%
\item the [[tensor product]] functor $\otimes \colon \mathcal{A} \times \mathcal{A} \longrightarrow \mathcal{A}$ is

\begin{enumerate}%
\item $k Mod$-[[enriched functor|enriched]] (i.e. $k$-linear);


\item [[exact functor|exact]]



\end{enumerate}
in both arguments;


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



\end{enumerate}
Such a $k$-tensor category is called \emph{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 $n \in \mathbb{N}$:

\begin{displaymath}
\itexarray{
    && \underset{i}{\oplus} E^{\otimes^{n_i}}
    \\
    && \downarrow
    \\
    X &\hookrightarrow& (\underset{i}{\oplus} E^{\otimes^{n_i}})/Q
  }
  \,.
\end{displaymath}
Such $E$ is called an \emph{$\otimes$-generator} for $\mathcal{A}$.

\end{defn}
(\hyperlink{Deligne02}{Deligne 02, 0.1})

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

\begin{displaymath}
\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
  \,.
\end{displaymath}
\end{defn}
(e.g. \hyperlink{EGNO15}{EGNO 15, def. 9.11.1})

\begin{defn}
\label{SchurFunctor}\hypertarget{SchurFunctor}{}
For $(\mathcal{A},\otimes)$ a $k$-[[tensor category]] as in def.\ref{TensorCategory}, 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

\begin{displaymath}
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)
\end{displaymath}
where

\begin{itemize}%
\item $S_n$ is the [[symmetric group]] on $n$ elements;


\item $V_\lambda$ is the [[irreducible representation]] of $S_n$ corresponding to $\lambda$;


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


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


\item the last expression just rewrites this as a [[group averaging]].



\end{itemize}
\end{defn}
(\hyperlink{Deligne02}{Deligne 02, 1.4})

\begin{prop}
\label{LenghtOfObjectIsBounded}\hypertarget{LenghtOfObjectIsBounded}{}
For a [[tensor category]] $\mathcal{A}$ the following are equivalent:

\begin{enumerate}%
\item the category has subexponential growth (def. \ref{SubexponentialGrowth}).


\item 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. \ref{SchurFunctor}, on $X$ vanishes: $S_\lambda(X) = 0$.



\end{enumerate}
\end{prop}
(\hyperlink{Deligne02}{Deligne 02, prop. 05})

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Pavel Etingof]], Shlomo Gelaki, Dmitri Nikshych, [[Victor Ostrik]], section 1.5 in \emph{Tensor categories}, Mathematical Surveys and Monographs, Volume 205, American Mathematical Society, 2015 (\href{http://www-math.mit.edu/~etingof/egnobookfinal.pdf
}{pdf})


\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Composition_series}{Composition series}}



\end{itemize}
The relation to [[Schur functors]] is discussed in

\begin{itemize}%
\item [[Pierre Deligne]], \emph{Cat\'e{}gorie Tensorielle}, Moscow Math. Journal 2 (2002) no. 2, 227-248. (\href{https://www.math.ias.edu/files/deligne/Tensorielles.pdf}{pdf})

\end{itemize}
For more on this see at \emph{[[Deligne's theorem on tensor categories]]}.

[[!redirects lengths of objects]]

[[!redirects Jordan-Hölder series]] [[!redirects Jordan-Holder series]]

[[!redirects Jordan-Hölder sequence]] [[!redirects Jordan-Holder sequence]]

[[!redirects object of finite length]] [[!redirects objects of finite length]]



\end{document}