May 20, 2019

Young Diagrams and Schur Functors

Posted by John Baez

What would you do if someone told you to invent something a lot like the natural numbers, but even cooler? A tough challenge!

I’d recommend ‘Young diagrams’.

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I gave a talk about Young diagrams yesterday at Math Connections — a conference organized by grad students here at U.C. Riverside. Check out my talk here:

Young diagrams are fundamental in group representation theory because they give ‘Schur functors’ — ways to turn one group representation into another, which apply in a completely general way to any representation.

Todd Trimble and I figured out a new way to think about this, which I explained briefly in my talk. Joe Moeller took notes and I polished them up a bit.

Posted at May 20, 2019 12:01 AM UTC

TrackBack URL for this Entry:   https://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/3112

Re: Young Diagrams and Schur Functors

Like the natural numbers, but cooler … my first thought would be PROPs, as expounded here and at https://graphicallinearalgebra.net (sorry; I don’t know how to make that a clickable link). On the other hand, as a representation theorist, I’m happy about anything that leads to representation theory. Is there any way to see PROPs as a special case of Young diagram?

Posted by: L Spice on May 20, 2019 7:03 PM | Permalink | Reply to this

Re: Young Diagrams and Schur Functors

No, but if you read my notes you’ll see Young diagrams are the “irreducible” objects in a very important symmetric monoidal category, the category of Schur functors. (This category actually has at least 5 important tensor products, but 4 of them are symmetric monoidal.) Since a PROP is a special sort of symmetric monoidal category, this should be enough to make you happy.

Posted by: John Baez on May 20, 2019 7:29 PM | Permalink | Reply to this

Christopher

Any of them distribute or otherwise interact interestingly?

Posted by: Christopher Andrew Upshaw on May 23, 2019 5:06 AM | Permalink | Reply to this

Re: Young Diagrams and Schur Functors

Schur functors interact in tons of interesting ways, like the Littlewood-Richardson rule (which says how the tensor product of Schur functors coming from Young diagrams can be written as a direct sum of Schur functors coming from Young diagrams), or plethysm (which says how the composite of Schur functors coming from Young diagrams can be written as a direct sum of Schur functors coming from Young diagrams). There’s a huge amount to say here… but most people will tell it to you in the language of ‘symmetric functions’, an adequate but decategorified version of the real thing.

Posted by: John Baez on May 24, 2019 1:04 AM | Permalink | Reply to this

Re: Young Diagrams and Schur Functors

Nice work! Schur was one smart cookie.

Young diagrams are more or less conjugacy classes in symmetric groups; more generally, conjugacy classes in any interesting group will probably turn out to be interesting. I’m not sure what conjugacy classes in the braid groups are; or, more precisely, how they’re related to links? As they say in `Private Eye’, I think we should be told…

Posted by: jackjohnson on May 21, 2019 9:57 PM | Permalink | Reply to this

Re: Young Diagrams and Schur Functors

I think what’s interesting about Young diagrams is that they are (at least) two things: not just conjugacy classes, but representations. These two sets of objects are in bijection for any finite group, but rarely in any nice way; other examples were sought, somewhat inconclusively, at MathOverflow.

Posted by: L Spice on May 23, 2019 8:04 PM | Permalink | Reply to this

Re: Young Diagrams and Schur Functors

You probably figure this out already, but it is worth a mention because of the pretty pictures: a braid $b$ is conjugate to another braid, $b'= g^{-1}bg$ precisely when $b$ and $b'$ have the same link as their respective closures. The closure is when you reconnect each strand of $b$ at the “bottom” of the braid to its original position at the “top.” So, in the closure of $b'$ the braid $g$ and its inverse can be pushed around the closure to cancel each other, leaving the same link as before!

Posted by: stefan on May 28, 2019 6:07 PM | Permalink | Reply to this

Re: Young Diagrams and Schur Functors

Is your construction related to Vershik-Okounkov approach? https://arxiv.org/abs/math/0503040

Posted by: ulyanick on June 6, 2019 12:53 PM | Permalink | Reply to this

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