## What is the name for a finite-group representation that is the sum of all the irreducibles (once)?

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I vaguely remember seeing a paper studying the concept of a totally multiplicity-one representation of a finite group, which concept, I recall, had a particular name, which I forget. What is this name, and is there a reference paper (for example, the one I might have been reading) where I can find out what is known about these representations?

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You might mean a Gelfand model. See Garge and Oesterle, and references degruyter.com/view/j/jgth.2010.13.issue-3/… – Junkie Feb 8 2012 at 3:31
It's going to be a bit hard to guess the paper, knowing only that it deals with models of finite groups... Are you interested in some specific kind of groups? (For example Lie type etc.) – Gjergji Zaimi Feb 8 2012 at 3:36
I believe Junkie has it (at least, the name, which rings a bell). I'm not wedded to the particular paper, if you have one you think is good. I believe the topic of that paper was a particular construction of a Gelfand model (i.e. an explicit vector space plus action), which may very well have been just for the symmetric groups. – Ryan Reich Feb 8 2012 at 3:41

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A model for a finite group $G$ is a representaton which contains each irreducible representation of $G$ with multplicity $1$. It is not clear this is what is being asked: maybe Ryan is asking for the name of a representation in which no irreducible representation ocurs with multiplicity greater than one, but not all irreducibles of $G$ occur (that's a multiplicity-free representation). This is in the semisimple context. R. Richardson showed that the symmetric groups have models which are sums of representations induced from 1-dimensional representations of centralizers of involutons. – Geoff Robinson Feb 8 2012 at 8:45

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I'm definitely asking for all irreducibles to have multiplicity exactly one. This may be clearer in the title, though. – Ryan Reich Feb 8 2012 at 17:21
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## 1 Answer

As others have commented, the most likely answer to the question in the header is Gelfand model, though you are apparently looking further for related literature. Probably the notion developed in the 1970s and 1980s in a series of papers by I.M. Gelfand and his collaborators. These are in Russian, but mostly published also in English translation journals and latter reprinted in the multi-volume collected works.

It's clear that the ideal notion of "model" of representations for a given group is a representation containing each irreducible representation exactly once as a constituent. But in some situations this is weakened to require only that "most" representations occur and/or that "most" of them have multiplicity just one. The groups studied usually are of Lie type (finite, compact, etc.) or perhaps closely related (symmetric groups, for example). And the results are varied, some more computable or usable than others. Finding a model for a specific group is a natural goal but difficult to reach.

MathSciNet and other databases provide good references and sometimes reviews. As a sample, here is a concise author summary which illustrates some of the typical motivation:

Gel′fand, I. M.; Zelevinski ̆ı, A. V. [Zelevinsky, Andrei], Models of representations of classical groups and their hidden symmetries. Akad. Nauk SSSR Inst. Prikl. Mat. Preprint 1984, no. 71, 26 pp.

Authors’ summary: “For all complex classical groups G we construct new realizations of the representation model of G, i.e., the direct sum of all irreducible algebraic finite-dimensional representations of G occurring with multiplicity one. These realizations have hidden symmetry: the action of the Lie algebra of G on them extends naturally to the action of a larger Lie (super-) algebra. The construction of hidden symmetries is based on a geometrical construction, similar to a twistor construction of Penrose.”

Another paper is part of a series in Russian by Bernstein-Gelfand-Gelfand:

Models of representations of compact Lie groups. (Russian) Funkcional. Anal. i Prilozˇen. 9 (1975), no. 4, 61–62.

[ADDED] Maybe I should emphasize that constructing an abstract model of representations for a group isn't by itself the goal, since it may be too big to provide further insight. As Junkie indicates (with reference to a paper that is also on the arXiv), symmetric groups and other finite Coxeter groups have been studied in this framework with partial success. For finite groups of Lie type, the ideas of Gelfand-Graev led to a simple construction which is "almost" a model of the ordinary characters but doesn't capture all of them. This is developed by Carter in his 1985 book Finite Groups of Lie Type, Chapter 8, and less completely by Digne-Michel in their 1991 text Representations of Finite Groups of Lie Type. Starting with any "regular" character of a maximal unipotent subgroup (the choices here don't matter), induction to the whole finite group yields a multiplicity-free character of large degree which has most other characters as constituents. Taken in isolation this is not so helpful, but combined with Deligne-Lusztig theory it leads to interesting results.

An underlying theme for groups of Lie type is that's it's fairly easy to construct big representations by induction from nice subgroups, but then it's not so easy to extract information about the irreducibles or multiplicities.

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