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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

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8 Comments & 0 Trackbacks

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 bb is conjugate to another braid, b′=g −1bgb'= g^{-1}bg precisely when bb and b′b' have the same link as their respective closures. The closure is when you reconnect each strand of bb at the “bottom” of the braid to its original position at the “top.” So, in the closure of b′b' the braid gg 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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