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## Generalized Frobenius-Schur numbers

Venue: | J. Algebra |

Citations: | 19 - 0 self |

### Citations

701 |
Atlas of finite groups
- Conway, Curtis, et al.
- 1985
(Show Context)
Citation Context ...hful permutation representation and an isomorphism H2 \Gamma ! S4. We take 2 to be the nontrivial one-dimensional character of H2. Labeling the irreducible characters as in the Atlas of Finite Groups =-=[5]-=-, we find that G1 = O/1 + O/2 + O/5, while G2 = O/3 + O/4 + O/6 + O/7 + O/8 + O/9, and so we have a generalized involution model in this case, too. A8 has no generalized involution model with o/ conju... |

335 |
Symmetric functions and Hall polynomials. Oxford Mathematical Monographs. The Clarendon Press Oxford
- Macdonald
- 1995
(Show Context)
Citation Context ...rst theorem asserts that if o/ g = tg\Gamma 1 then fflo/ (ss) = 1 for all irreducible representations. As Howlett and Zworestine noted, Theorem 2 shows the equivalence in a transparent way. Macdonald =-=[20]-=- removed the assumption that q is odd. Fulman and Guralnick [11] in a remarkable paper give explicit formulas for O/M at all conjugacy classes of G = GL(n; Fq), and striking interpretations of these r... |

220 |
Lectures on Chevalley groups
- Steinberg
- 1968
(Show Context)
Citation Context ... they are conjugate. \LambdasThe following fact is well-known in the special case where o/ = 1. The case where z = 1 is in Howlett and Zworestine [15]. Motivated by examples of Prasad [21], Steinberg =-=[25]-=- and Gow [12], and Proposition 3 below, we generalize this by using a central element z (which must be of order two). If (ss; V ) is an irreducible representation of G, let !ss : Z(G) \Gamma ! C\Theta... |

175 |
Automorphic Forms and Representations
- Bump
- 2008
(Show Context)
Citation Context ...f By the geometric form of Mackey's theorem the endomorphism ring of G consists of the convolution ring of functions \Deltason G satisfying \Delta (hgh0) = (h) \Delta (g) (h0) for h, h0 2 H (see Bump =-=[4]-=-, Proposition 4.1.2). Defining '\Delta (g) = \Delta ('g), since ' is anticommutative, '(\Delta 1 \Lambdas\Delta 2) = '\Delta 2 \Lambdas'\Delta 1. The assumption that every double coset is '-invariant ... |

30 |
Characters of finite groups
- Feit
- 1967
(Show Context)
Citation Context ... values to be nonnegative, so strong that one might wonder whether this is always true. Solomon [23] proved that the row sums of the character table are nonnegative. This result is reproduced in Feit =-=[8]-=- p. 34, who remarks that Solomon and Thompson had pointed out that the column sums of the character table (which are the values of O/M) can be negative. Still, we had to look fairly hard to find an ex... |

22 |
An explicit model for the complex representations
- Inglis, Richardson, et al.
- 1990
(Show Context)
Citation Context ...nvolution model . (For the purpose of this definition, the identity element of G is considered an involution.) A good example is for Sn, where the model was constructed by Inglis, Richardson and Saxl =-=[16]-=-, and independently (details unpublished) by Klyachko. Baddeley [1] shows that most but not all Weyl groups have involution models. If o/ 6= 1, it is natural to call a model for G arising from a set o... |

17 |
representations of the finite orthogonal and symplectic groups of odd characteristic
- Gow, “Real
- 1985
(Show Context)
Citation Context ...jugate. \LambdasThe following fact is well-known in the special case where o/ = 1. The case where z = 1 is in Howlett and Zworestine [15]. Motivated by examples of Prasad [21], Steinberg [25] and Gow =-=[12]-=-, and Proposition 3 below, we generalize this by using a central element z (which must be of order two). If (ss; V ) is an irreducible representation of G, let !ss : Z(G) \Gamma ! C\Thetasbe the centr... |

17 |
On a theorem of Frobenius
- Hall
- 1936
(Show Context)
Citation Context ...re ae is a primitive cube root of unity. We find that fflo/ (O/) = 1 + 2ae. Let us give an application. If g 2 G let M (g) be the number of solutions to the equation N (x) = g with x 2 G. Philip Hall =-=[14]-=-, generalizing a theorem of Frobenius, proved that M (g) is divisible by the greatest common divisor of r and the order of the centralizer of g. An interesting integer valued function on the group--is... |

17 |
A twisted version of the Frobenius-Schur indicator and multiplicity-free permutation representations
- Kawanaka, Matsuyama
- 1990
(Show Context)
Citation Context ...ial, then fflo/ (O/) is (1=jGj) P O/(g2), and such sums were considered by Frobenius and Schur [10]. If r = 2 and o/ is not assumed to be trivial, these sums were considered by Kawanaka and Matsuyama =-=[18]-=-. The work of Kawanaka and Matsuyama has many interesting associations, for example to the notion of a generalized involution model which we discuss. We will consider a couple of different generalizat... |

16 |
On the self-dual representations of finite groups of Lie type
- Prasad
- 1998
(Show Context)
Citation Context ...o/ g\Gamma 1, so they are conjugate. \LambdasThe following fact is well-known in the special case where o/ = 1. The case where z = 1 is in Howlett and Zworestine [15]. Motivated by examples of Prasad =-=[21]-=-, Steinberg [25] and Gow [12], and Proposition 3 below, we generalize this by using a central element z (which must be of order two). If (ss; V ) is an irreducible representation of G, let !ss : Z(G) ... |

16 |
Geometrical Gelfand models, tensor quotients and Weil representations
- Soto-Andrade
- 1987
(Show Context)
Citation Context ... therefore impossible. \LambdasOn the other hand, a variant of this setup does produce a model for PGL(2; Fq), where q is odd. The following construction is similar to one 10sproposed by Soto-Andrade =-=[24]-=-. However Soto-Andrade makes exceptions for the one-dimensional representations, and we do not. In PGL(2; Fq), let H1 be the normalizer of the diagonal torus, let H2 be the normalizer of an anisotropi... |

15 | Conjugacy class properties of the extension of GL(n; q) generated by the inverse transpose involution
- Fulman, Guralnick
(Show Context)
Citation Context ...1 for all irreducible representations. As Howlett and Zworestine noted, Theorem 2 shows the equivalence in a transparent way. Macdonald [20] removed the assumption that q is odd. Fulman and Guralnick =-=[11]-=- in a remarkable paper give explicit formulas for O/M at all conjugacy classes of G = GL(n; Fq), and striking interpretations of these results. When n = 2, in addition to the involution o/ , there is ... |

14 |
Two remarks on irreducible characters of finite general linear groups
- Shintani
- 1976
(Show Context)
Citation Context ...on G by conjugation according to the automorphism o/ : tgt\Gamma 1 = o/ g for g 2 G. Let U = V \Omega \Deltas\Deltas\Delta \Omega V . We will give it a ^G module structure which was noted by Shintani =-=[22]-=-, Lemma 1-4. Specifically, let G act by the representation \Pi (g) = ss(g) \Omegasss(o/ g) \Omegas: : : \Omegasss(o/r\Gamma 1g); which we extend to ^G by letting \Pi (t)(v1 \Omegas: : : \Omegasvr) = v... |

13 |
Properties of the characters of the finite general linear group related to the transpose-inverse involution
- Gow
- 1983
(Show Context)
Citation Context ... Thus if the equivalent conditions of Theorem 2 are satisfied, we have a suitable combinatorial interpretation of O/M: Proof This is immediate from Proposition 1. \LambdasLet us recall results of Gow =-=[13]-=- and Klyachko [19]. Independently, Gow and Klyachko proved two theorems, and proved the equivalence of them. Let G be GL(n; Fq), and let (ss; V ) be an irreducible representation. Theorem 4 (Gow, Klya... |

11 |
An explicit model for the complex representations of the finite general linear groups
- Inglis, Saxl
- 1991
(Show Context)
Citation Context ...Sn\Gamma 2r, where W (Cr) is the Weyl group of Sp(2r) realized as a subgroup of S2r, and these groups are precisely the centralizers of involutions in Sn used in the involution model. Inglis and Saxl =-=[17]-=-, being aware of this connection, and gave a variant of Klyachko's model in which this relationship between Klachko's GL(n) model and the models for the symmetric group is made clarified. But despite ... |

10 |
Pioneers of representation theory: Frobenius, Burnside, Schur and Brauer. History of Mathematics, vol 15. Published by Amer
- Curtis
- 1999
(Show Context)
Citation Context ...n of a famous theorem of Shintani on the lifting of characters for GL(n) over a finite field. This paper touches on different works of Frobenius and Schur. An invaluable guide to their work is Curtis =-=[6]-=-. We would like to thank Persi Diaconis, Ryan Vinroot, editor Jan Saxl and the referee for helpful comments. This work was supported in part by NSF grant DMS-9970841. 1 Theorems of Frobenius-Schur and... |

9 |
Models and Involution Models for Wreath Products and certain Weyl Groups
- Baddeley
- 1991
(Show Context)
Citation Context ...element of G is considered an involution.) A good example is for Sn, where the model was constructed by Inglis, Richardson and Saxl [16], and independently (details unpublished) by Klyachko. Baddeley =-=[1]-=- shows that most but not all Weyl groups have involution models. If o/ 6= 1, it is natural to call a model for G arising from a set of stabilizers for the action of G on X by twisted conjugacy a gener... |

8 | On Klyachko’s model for the representations of finite general linear groups
- Howlett, Zworestine
- 1998
(Show Context)
Citation Context ...aracters take the same value on g and o/ g\Gamma 1, so they are conjugate. \LambdasThe following fact is well-known in the special case where o/ = 1. The case where z = 1 is in Howlett and Zworestine =-=[15]-=-. Motivated by examples of Prasad [21], Steinberg [25] and Gow [12], and Proposition 3 below, we generalize this by using a central element z (which must be of order two). If (ss; V ) is an irreducibl... |

7 |
Models for complex representations of the groups GL(n,q
- Klyachko
- 1983
(Show Context)
Citation Context ...alent conditions of Theorem 2 are satisfied, we have a suitable combinatorial interpretation of O/M: Proof This is immediate from Proposition 1. \LambdasLet us recall results of Gow [13] and Klyachko =-=[19]-=-. Independently, Gow and Klyachko proved two theorems, and proved the equivalence of them. Let G be GL(n; Fq), and let (ss; V ) be an irreducible representation. Theorem 4 (Gow, Klyachko) Let (ss; V )... |

4 |
Models of representations of compact Lie groups
- Bernstein, Gelfand, et al.
- 1975
(Show Context)
Citation Context ...tation ss is an embedding of ss in a multiplicity free induced representation, typically induced from a onedimensional representation of a subgroup of G. The project of Bernstein, Gelfand and Gelfand =-=[3]-=- is to find several subgroups H1; \Deltas\Deltas\Deltas; Hn of G and characters i (typically one dimensional) of Hi such that M i IndGHi (i) ,= M: (7) Of course this means that IndGHi (i) is multiplic... |

3 |
descent and L functions on Deligne-Lusztig varieties, The Arcata Conference on Representations of Finite Groups
- Digne, Shintani
- 1986
(Show Context)
Citation Context ... an automorphism of G. The number of twisted conjugacy classes of G with respect to o/ equals the number of o/ -invariant irreducible representations of G. This is implied by Proposition 1.2 of Digne =-=[7]-=-, which is stated without proof; it was most likely known to Shintani. We give a simple counting argument. Proof There are two permutation actions of ho/ i which we may consider: the action on the con... |

2 |
Finite groups in which every element is conjugate to its inverse
- Berggren
- 1969
(Show Context)
Citation Context ...ly cases where there exists an involution o/ of An such that g is conjugate to o/ g\Gamma 1 for all g 2 An. The first statement, about An when n is conjugate to its inverse is also proved in Berggren =-=[2]-=-. Proof For any g 2 An, g is conjugate to its inverse in Sn. If the centralizer CSn(g) is not contained in An then it follows that g , g\Gamma 1 in An, too. It is easy to see that if CSn(g) ` An then ... |

2 |
Uber die rellen Darstellungen der endlichen
- Frobenius, Schur
- 1906
(Show Context)
Citation Context ...ers O/ of G. (The proof is a straightforward adaptation of Proposition 1 below.) If r = 2, and o/ is trivial, then fflo/ (O/) is (1=jGj) P O/(g2), and such sums were considered by Frobenius and Schur =-=[10]-=-. If r = 2 and o/ is not assumed to be trivial, these sums were considered by Kawanaka and Matsuyama [18]. The work of Kawanaka and Matsuyama has many interesting associations, for example to the noti... |

2 |
On the sum of the elements in the character table of a finite group
- Solomon
- 1961
(Show Context)
Citation Context ...( _Q=Q) permutes the constituents of M, they are rational integers. There is a strong tendency for these values to be nonnegative, so strong that one might wonder whether this is always true. Solomon =-=[23]-=- proved that the row sums of the character table are nonnegative. This result is reproduced in Feit [8] p. 34, who remarks that Solomon and Thompson had pointed out that the column sums of the charact... |

1 |
Uber die charakterisischen Einheiten der symmetrischen
- Frobenius
- 1903
(Show Context)
Citation Context ...e type *. For example if * = (3; 2; 2) then we can take c* = (123)(45)(67): A fundamental formula, known to Frobenius, asserts that if g 2 U (n), then Y tr(g*i) = X ` O/`(c*)j`(g): (14) See Frobenius =-=[9]-=-, (1.) of Section 4. Frobenius' result is stated in terms of symmetric polynomials, but it can be translated into a statement about characters of irreducible representations of unitary groups. His m i... |