Do representations uniquely determine groups?
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From: metroplex021 on 13 Jan 2010 21:55 I recently read (in 'Symmetries in Physics' by McVoy) that all groups "possess a definite, unambiguous (and to some extent distinctive) set of multiplets". Does anyone know what the theorem is that establishes the extent to which a knowledge of the representations suffices to determine the group - or just a good place to go read about this...
From: Maarten Bergvelt on 14 Jan 2010 09:27 On 2010-01-14, metroplex021 <fourthinternational(a)googlemail.com> wrote: > I recently read (in 'Symmetries in Physics' by McVoy) that all groups > "possess a definite, unambiguous (and to some extent distinctive) set > of multiplets". Does anyone know what the theorem is that establishes > the extent to which a knowledge of the representations suffices to > determine the group - or just a good place to go read about this... This is called the Tannakian philosophy: from the category of representations you can reconstruct (in many cases, I am not quite sure what the conditions are) the group. See for instance the Math Overflow question: http://mathoverflow.net/questions/3446/tannakian-formalism Any finite dimensional representation of a group has a character; the characters are not enough to determine the group, see MR1084537 (92b:01048) Shafarevich, I. R.(2-AOS) Abelian and nonabelian mathematics. Translated from the Russian by Smilka Zdravkovska. Math. Intelligencer 13 (1991), no. 1, 67--75. (This is anyway a great piece of expository writing!) -- Maarten Bergvelt
From: A on 14 Jan 2010 19:15 On Jan 14, 9:27 am, Maarten Bergvelt <be...(a)math.uiuc.edu> wrote: > On 2010-01-14, metroplex021 <fourthinternatio...(a)googlemail.com> wrote: > > > I recently read (in 'Symmetries in Physics' by McVoy) that all groups > > "possess a definite, unambiguous (and to some extent distinctive) set > > of multiplets". Does anyone know what the theorem is that establishes > > the extent to which a knowledge of the representations suffices to > > determine the group - or just a good place to go read about this... > > This is called the Tannakian philosophy: from the category of > representations you can reconstruct (in many cases, I am not quite > sure what the conditions are) the group. > I think the relevant phrase to Google here is "Tannaka-Krein duality." > > See for instance the Math Overflow question:http://mathoverflow.net/questions/3446/tannakian-formalism > > Any finite dimensional representation of a group has a character; the > characters are not enough to determine the group, see > > MR1084537 (92b:01048) > Shafarevich, I. R.(2-AOS) > Abelian and nonabelian mathematics. > Translated from the Russian by Smilka Zdravkovska. > Math. Intelligencer 13 (1991), no. 1, 67--75. > > (This is anyway a great piece of expository writing!) > -- > Maarten Bergvelt
From: metroplex021 on 14 Jan 2010 23:33 On Jan 15, 12:15 am, A <anonymous.rubbert...(a)yahoo.com> wrote: > On Jan 14, 9:27 am, Maarten Bergvelt <be...(a)math.uiuc.edu> wrote: > > > On 2010-01-14, metroplex021 <fourthinternatio...(a)googlemail.com> wrote: > > > > I recently read (in 'Symmetries in Physics' by McVoy) that all groups > > > "possess a definite, unambiguous (and to some extent distinctive) set > > > of multiplets". Does anyone know what the theorem is that establishes > > > the extent to which a knowledge of the representations suffices to > > > determine the group - or just a good place to go read about this... > > > This is called the Tannakian philosophy: from the category of > > representations you can reconstruct (in many cases, I am not quite > > sure what the conditions are) the group. > > I think the relevant phrase to Google here is "Tannaka-Krein duality." > > > > > See for instance the Math Overflow question:http://mathoverflow.net/questions/3446/tannakian-formalism > > > Any finite dimensional representation of a group has a character; the > > characters are not enough to determine the group, see > > > MR1084537 (92b:01048) > > Shafarevich, I. R.(2-AOS) > > Abelian and nonabelian mathematics. > > Translated from the Russian by Smilka Zdravkovska. > > Math. Intelligencer 13 (1991), no. 1, 67--75. > > > (This is anyway a great piece of expository writing!) > > -- > > Maarten Bergvelt > > Thanks folks. I looked up Tannaka-Krein duality, which does indeed seem to establish that a compact group can be reconstructed from the category of its unitary representations. This is definitely interesting. However, it has just struck me that the 'multiplets' referred to in the quote from McVoy correspond to *irreducible* representations. These don't exhaust the unitary reps - and they don't (to my knowledge) form a category by themselves at all. So I guess my question is really whether a (compact) group can be reconstructed from a knowledge of its irreducible representations, hence from its fundamental representation. (I initially thought that they could be was implicit in the very word 'representation' - but showing it's the case is proving very difficult!) But thanks for the hook-up with the T-K duality - it is useful.
From: José Carlos Santos on 15 Jan 2010 02:32
On 15-01-2010 4:33, metroplex021 wrote: >>>> I recently read (in 'Symmetries in Physics' by McVoy) that all groups >>>> "possess a definite, unambiguous (and to some extent distinctive) set >>>> of multiplets". Does anyone know what the theorem is that establishes >>>> the extent to which a knowledge of the representations suffices to >>>> determine the group - or just a good place to go read about this... >> >>> This is called the Tannakian philosophy: from the category of >>> representations you can reconstruct (in many cases, I am not quite >>> sure what the conditions are) the group. >> >> I think the relevant phrase to Google here is "Tannaka-Krein duality." >> >> >> >>> See for instance the Math Overflow question:http://mathoverflow.net/questions/3446/tannakian-formalism >> >>> Any finite dimensional representation of a group has a character; the >>> characters are not enough to determine the group, see >> >>> MR1084537 (92b:01048) >>> Shafarevich, I. R.(2-AOS) >>> Abelian and nonabelian mathematics. >>> Translated from the Russian by Smilka Zdravkovska. >>> Math. Intelligencer 13 (1991), no. 1, 67--75. >> >>> (This is anyway a great piece of expository writing!) > > Thanks folks. I looked up Tannaka-Krein duality, which does indeed > seem to establish that a compact group can be reconstructed from the > category of its unitary representations. This is definitely > interesting. However, it has just struck me that the 'multiplets' > referred to in the quote from McVoy correspond to *irreducible* > representations. These don't exhaust the unitary reps - and they > don't (to my knowledge) form a category by themselves at all. Actually, if K is a compact group then for *any* continuous action of K on C^n (irreducible or not) there is an inner product on C^n which turns it into an unitary representation. Besides, I am not sure about what you mean when you say that irreducible representations don't "form a category by themselves", but that seems natural to me, since irreducible representations are meant to be the building blocks of all representations. Best regards, Jose Carlos Santos |