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Does software exists for calculating with planar algebras or group rings? It could be part of Mathematica or be an extension of Python or Java or C. What would go into designing such a data-type anyway? One would have to define a special class describing elements of your algebra and overload "+" and "x" so it felt natural.

This request comes about because I am too lazy to write out all 16 terms in the young symmetrizer for the partition (2,2) and show that representation is two-dimensional space. This will be even more unwieldy for larger partitions.

It would also be difficult to do computations with the Templerley-Lieb algebras without software.

An alternative question is: are there efficient ways to compute with Specht modules or the Temperley-Lieb algebras by hand?

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  • $\begingroup$ The Temperley-Lieb algebra is pretty easy to compute with by hand! $\endgroup$
    – Sammy Black
    Commented Mar 27, 2010 at 23:45
  • $\begingroup$ @Sammy, it depends what you're doing. I was recently hoping to find 'low weight vectors' (i.e. elements killed by all caps) in $TL_d \otimes TL_e$, where $d$ and $e$ are both 'special values'. I was stymied by the lack of a good library for Temperley-Lieb arithmetic. $\endgroup$
    – Kim Morrison
    Commented Mar 28, 2010 at 0:31
  • $\begingroup$ Generally -- anyone interested in writing programs to do calculations in planar algebras should talk to me, Noah Snyder and Emily Peters! We have lots of enthusiasm and half-baked ideas. $\endgroup$
    – Kim Morrison
    Commented Mar 28, 2010 at 0:32

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Sage can do some things with group algebras, in particular, with group algebras for symmetric groups, but it doesn't seem to have anything about planar algebras. For example:

S = SymmetricGroupAlgebra(ZZ, 3) 
# ZZ, the integers, is the coefficient ring
# "3" means the symmetric group on 3 letters
a = S([2,1,3])  # turn the permutation [2,1,3] into an element of S
b = S([3,1,2])
(2*a + b)^2

prints out

4*[1, 2, 3] + 2*[1, 3, 2] + [2, 3, 1] + 2*[3, 2, 1]

If you'd started with a different coefficient ring:

S = SymmetricGroupAlgebra(GF(3), 3)

then the output from the above would be

[1, 2, 3] + 2*[1, 3, 2] + [2, 3, 1] + 2*[3, 2, 1]

You can also do computations with other group algebras for other groups, but symmetric group algebras seem to be a bit better developed.

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Both Magma and GAP contain interfaces for doing computations with group algebras.

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  • $\begingroup$ Magma is not free. Maybe my school has it. $\endgroup$
    – john mangual
    Commented Mar 28, 2010 at 13:26
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    $\begingroup$ While it is unfortunately true that MAGMA is not free, it is in my mind the best tool for computing with representation theory of finite dimensional algebras (and especially group algebras). Many institutions have licenses but you have to run it through ssh. The biggest advantage to my mind is that MAGMA authors were aware of basic notions in category theory, and so when you work with magma you have all the maps connecting your various objects (e.g. quotients, inclusions, etc.) built in. For coding many things, this is indispensably convenient. $\endgroup$
    – David Jordan
    Commented Mar 28, 2010 at 14:33
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If there is software for planar algebras I would interested in hearing about it.

For working in the Hecke algebra see my answer to Do Jones-Wenzl idempotents lift to anything interesting in the Hecke algebra? and for the one specific question you ask see my answer to Sammy's previous question Does the super Temperley-Lieb algebra have a Z-form?

You should be able to do what you want by hand. If you can't then can you say what the problem is.

Also I don't see the connection between the two problems. How would planar algebra software help you with these algebras?

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