## Categorification and Schur functors

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Are Schur functors and categorification somehow related ?

If yes, probably looking on Schur functors (which I know) one can illustrate on this example why "categorification" (which I do not know) is so important/popular now ? (I am intersted to learn somehting about "categorification", but I would prefer to have some "good entering point", meaning to relate it to what I know).

Schur functors - are some functors from category of vector spaces to itself. http://en.wikipedia.org/wiki/Schur_functor For example take vector space and send it to $S^n(V)$ n-th symmetric power (can be antisymmetric).

"Categorification" - briefly looking at MO discussions about it and abstracts of some papers, I got an impression that it is about realizing certain algebras as functors of some categories. For example abstact of Khovanov's lectures http://arxiv.org/abs/1008.5084 contains the following sentence " diagrammatic categorification of positive halves of quantum groups". Is my understanding correct ?

PS

http://mathoverflow.net/questions/4841/what-precisely-is-categorification

http://mathoverflow.net/questions/89001/algebraic-relations-between-schur-functors

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1
The nlab article on Schur functors is highly recommended ncatlab.org/nlab/show/Schur+functor. The message is that the category of Schur functors is just End(U), where U is the forgetful functor from cocomplete (or at least Cauchy-complete) tensor categories to categories. – Martin Brandenburg Feb 22 2012 at 13:59

## 1 Answer

Categorification can be thought of as the process of replacing equalities with isomorphisms (in some category). A basic example is replacing a numerical combinatorial identity such as $$2^n = \sum_{k=0}^n {n \choose k}$$

with a bijection between two sets (an isomorphism in $\text{Set}$); in this case, between the set of subsets of ${ 1, 2, ... n }$ and the disjoint union of the sets of subsets of size $k$ over all $k$.

As another example, the category $\text{FinSet}$ of finite sets itself categorifies the rng $\mathbb{N}_{\ge 0}$ of non-negative integers; the coproduct categorifies addition and the product categorifies multiplication. (Also taking Hom-sets categorifies exponentiation.) In this way we replace equalities such as $1 + 1 = 2$ with isomorphisms $1 \sqcup 1 \cong 2$ (where $1$ is the set with one element, $2$ is the set with two elements, and $\sqcup$ is the coproduct).

Schur functors categorify the theory of symmetric functions (specifically of Schur functions). Taking the direct sum of Schur functors categorifies addition of symmetric functions, taking the tensor product categorifies multiplication of symmetric functions, and composing categorifies plethysm. So here we replace equalities between expressions involving symmetric functions with natural isomorphisms between expressions involving Schur functors.

As far as the linked arXiv paper goes, the idea is to replace a ring $A$ by a category $C$ with direct sums and tensor products such that the latter distributes over the former, such as the category of $(R, R)$-bimodules for some ring $R$. Equality of elements in $A$ is then replaced by isomorphism between objects in $C$. Taking the Grothendieck group of $C$ (a form of decategorification) should then recover the original ring.

One reason to attempt to do this is that the category $C$ may have a distinguished set of objects that provides a distinguished set of generators for $A$. For example, the Hecke algebra $H_n$ of $S_n$ turns out to be the Grothendieck group of the category of Soergel bimodules. Among these we can distinguish the indecomposable modules, and these turn out to give the Kazhdan-Lusztig basis of $H_n$.

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@Qiauchu Yuan Thank You for the answer ! What do you mean by "composing" symmetric functions ? – Alexander Chervov Feb 21 2012 at 18:30
Added phrase: "For example abstact of Khovanov's lectures arxiv.org/abs/1008.5084 contains the following senteces " diagrammatic categorification of positive halves of quantum groups"." Can you comment on it ? It does not seems like "identities" categorification. – Alexander Chervov Feb 21 2012 at 18:34

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@Alexander: Schur functors are the things that can be composed. Decategorifying this construction gives rise to plethysm of symmetric functions. – Qiaochu Yuan Feb 21 2012 at 18:38
@Alexander: it is. See Proposition 2 on p. 23. – Qiaochu Yuan Feb 21 2012 at 18:43
@Qiauchu Can plethysm be described solely in terms of symmetric functions and what does it mean from this point of view ? – Alexander Chervov Feb 22 2012 at 5:48
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