I. Schur, Uber die Darstellung der symmetrischen und der alternirenden Gruppe durch gebrochene lineare Substitutionen, J. Reine Angew. Math. 139 (1911), 155-- 250.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Set r = i h j 0 j ( 0 ) Then h i(x (e m ) 2 r h 0 i(1) 1 ] 1) t ] 1) Remark. Since (n) n s(n) we have ( t s( j ) s(m) For a reduction to the analogues of Young subgroups we have to collect some results on reduced Cli ord products (see [H] MiO] [S]) Theorem 2.4 Let i ; i be spin characters of S a i , i = 1; k. Then the reduced Cli ord products = c i and = c i have the following properties. i) is n.s.a. if and only if t = jfi j i n.s.a.gj is odd. ii) Let be a partition of a i , i = 1; k, ....

I. Schur:  Uber die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen. J. reine ang. Math. 39 (1911) 155-250 (ges. Abhandlungen 1, 346-441, Springer-Verlag 1973)

....We use the following vertex operators on Sym: s 1 DP2 s 2 Gamma Delta Delta Delta ; 5:22) e 1 DP2 e 2 Gamma Delta Delta Delta ; 5:23) 1 Gamma D e1 P 1 D e2 P 2 Gamma Delta Delta Delta : 5:24) We refer to [LP1, p. 24] for the definitions of Schur P functions P I [S]. In loc. cit. the reader can also find a definition of Q functions Q I [LLT2] used in the following proposition: Proposition 25. Let I be a strict partition. We have the following identities of symmetric functions in Sym: e Q I U e Q I ; I) even (5:25) I) even ....

I. Schur, Uber die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen, J. reine angew. Math. 139 (1911), 155-250.

....process) of Young diagrams. In the present paper, we intend to do the same for the non trivial central extensions of Sn generally denoted by e Sn : 1 Z e Sn Sn 1; where Z is of order two (see the de nition in x1.1 below) The irreducible characters of e Sn where determined by I. Schur (see [10]) the ones that do not have Z in their kernel are indexed by partitions of n without repetition. The analogue of Nakayama conjectures for e Sn was rst stated by A.O. Morris (see [6] and checked only recently by J.H. Humphreys (see [5] using previous papers on blocks of projective ....

....G, the inclusion (P 1 ; e 1 ) P 2 ; e 2 ) holds if, and only if, P 1 P 2 and the two subpairs are both included in a third one (for instance a maximal one) The paper is organized as follows. Part 1 recalls the basic facts on e Sn and its irreducible characters that go back to Schur s work ([10],1911) along with Morris theory of p bars ( 6] 1965) which gives the recursive way to compute character values. Part 2 is devoted to the necessary background on local block theory, we follow [1] quite closely. Part 3 is divided as follows. In section 3.1, we relate the centralizers of p subgroups ....

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. I. Schur,  Uber die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen, J. Reine Angew. Math., 139(1911), 155-250. 22

....determinants [12] to the analogous rules for Pfaffians. Acknowledgements. Discussions with Lyle Ramshaw helped greatly to clarify my proof of (1.0) Paul Algoet kindly corrected several typographical errors in my preprint. Alain Lascoux referred me to [10] and [12] John Stembridge told me about [22], and an anonymous referee called my attention to [8] I also thank the editors for their patience. ....

J. Schur, " Uber die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen," Journal fur die reine und angewandte Mathematik 139 (1911), 155--250. Reprinted in Issai Schur, Gesammelte Abhandlungen 1 (1973), 346--441.

....intertwining operator Ms. x M x over the algebra H(q) As usual, we write T Tk . Tkp for any choice of reduced decomposition s = sk skp in the standard generators Sl, s l. This approach gives an explicit formula for the element E different from that given in [4] cf. 13] It was Schur [17] who discovered non trivial central extensions of the symmetric group S. In other words, the group S admits projective representations which cannot be reduced to linear ones. The analogues of the Young symmetrizers (1.1) for these representations were constructed by the second author in [13] using ....

....scalars. These central elements act in U by the same scalars. These scalars distinguish the modules U with different strict partitions h n. The G[ q) module U with d 1 cannot be absolutely irreducible by Proposition 6.3. But Theorem 6.2(a) and Corollary 6. 4 along with the classical result of [17] show that which is exactly the dimension of the algebra G(q) over the field F, see Proposition 2.1. So if U7 is a proper submodule in U K or if the G(q) module U with d = 0 is not absolutely irreducible, we get a contradiction with Propositions 2.2 and 6.5(b) In the course of the proof of ....

I. SCHUR, 'Uber die Darstellung der symmetrischen und der alternierenden Gruppe dutch gebrochene lineare Substitutionen', J. Reite Atgew. Math. 139 (1911) 155 250.

....4) 8 j 0 Otherwise D is irreducible. The proof of this theorem depends on the reduction modulo two of spin representations. We first recall the basic definitions and notations, and then we state Theorem 7 of [2] as our Theorem 1.2. The basic references for spin representations are Schur [12] and Morris [7, 8, 9, 10] Let Gamma n be one of the two isoclinic proper double covers of Sigma n (it doesn t matter which although we shall make an explicit choice later) Corresponding to a partition = 1 ; s ) of n into s unequal parts, we have either one faithful irreducible ....

....of transpositions have order two. Any representation of one may be made into a representation of the other by multiplying by i all the matrices corresponding to elements not in the subgroup of index two. References for the representation theory of Gamma n are Morris [7, 8, 9, 10] and Schur [12], and for an explanation of isoclinism see Beyl and Tappe [3] 4 Description of the Clifford lattice Delta and its restriction to Gamma n If n is odd, let Xn 1 be a space of one larger dimension than X , with basis e 1 ; e n 1 and the usual quadratic form. Then C(X) C(Xn 1 ) and the ....

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I. Schur. Uber die Darstellung der symmetrischen und der alternierenden Gruppe durch lineare Substitutionen. J. Reine Angew. Math. 139, 155--250 (1910).

....Calculus and the class of the diagonal Let us first recall the following result on Lagrangian and orthogonal Schubert Calculus from [P1,2] We need two families of polynomials in the Chern classes of a vector bundle E over a smooth variety X. Their construction is inspired by I. Schur s paper [S]. The both families will be indexed by partitions (i.e. by sequences I = i 1 : i k 0) of integers) Set, in the Chow ring A (X) of X, for i j 0: e Q i;j E : c i E Delta c j E 2 c i p E Delta c j Gammap E; so, in particular e Q i E : e Q i;0 E = c i E for i 0. In ....

.... Delta e P ae n Gamma1 rI where GF = G Theta X F (resp. GF = G Theta X F ) using Propositions 2.7 and 3.5. 4. e Q polynomials and their properties In this section we define a family of symmetric polynomials modelled on Schur s Q polynomials. In Schur s Pfaffian definition (see [S]) we replace Q i by e i the i th elementary symmetric polynomial. After this modification one gets an interesting family of symmetric polynomials e Q I (indexed by all partitions) whose properties are studied in this section and then applied in the next ones. It turns out that e Q I is the ....

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I. Schur, Uber die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen, J. reine angew. Math. 139 (1911), 155-250.

....est strictement contenue dans l espace lin eaire Xi des fonctions quasi sym etrique gauchis. Puis nous montrons que Xi est une coalg ebre et calculons les dimensions des composantes de degr ee n. 1. Introduction Schur Q functions first arose in the study of projective representations of S n [8]. Since then they have appeared in variety of contexts including the representations of Lie superalgebras [9] and cohomology classes dual to Schubert cycles in isotropic Grassmanians [4, 7] While studying the duality between skew Schur P and Q functions and their connection to the Schubert ....

I. Schur, Uber die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen, J. reine angew. Math. , 139 (1911) 155--250.

....que est strictement contenue dans l espace lin eaire des fonctions quasi sym etrique gauchis. Puis nous montrons que est une coalg ebre et calculons les dimensions des composantes de degr ee n. 1. Introduction Schur Q functions rst arose in the study of projective representations of S n [8]. Since then they have appeared in variety of contexts including the representations of Lie superalgebras [9] and cohomology classes dual to Schubert cycles in isotropic Grassmanians [4, 7] While studying the duality between skew Schur P and Q functions and their connection to the Schubert ....

I. Schur,  Uber die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen, J. reine angew. Math. , 139 (1911) 155-250.

....Introduction In this article, we determine the irreducible projective representations of the symmetric group S d and the alternating group A d over an algebraically closed eld of characteristic p 6= 2. These matters are well understood in the case p = 0, thanks to the fundamental work of Schur [19] in 1911, as well as the much more recent work of Nazarov [16, 17] and others. So the focus here is primarily on the case of positive characteristic, where surprisingly little is known as yet. In particular, we obtain a natural combinatorial labelling of the irreducibles in terms of a certain set ....

I. Schur,  Uber die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen, J. Reine Angew. Math. 139 (1911), 155{ 250.

....partitions from the first family, but there is an explicit formula, rather than a generating function, for their values on the second (see [Mor 77] for details) If has only odd parts, the values of all three characters are the same and we will denote this common value by i . Theorem 7.2. 1 ( Scu 11] If j= n then Q (x) 1 n X n 2 d l( l( 2 e c i p (x) where the sum is over all partitions with only odd parts, and d Deltae is the round up or ceiling function. 2 Corollary 7.1.5 also has a projective analog. Theorem 7.2.2 (x 1 x 2 Delta Delta Delta x n ) k ....

I. Schur, Uber die Darstellung der symmetrischen und der alterneinden Gruppe durch gebrochene lineare Substitutionen, J. Reine Angew. Math, 139 (1911), 155-250.

.... are due to Hiller and Boe [6] whose proof used the Chevalley formula [2] Another proof, using the Leibnitz formula for symplectic and orthogonal divided differences, was given by Pragacz and Ratajski [14] These formulas also arise in the theory of projective representations of symmetric groups [15, 9] as product formulas for Schur P and Q functions, and were first proven in this context by Morris [11] The connection of Schur P and Q functions to geometry was established by Pragacz [12] see also [13] In Section 1, we give the basic definitions and state the Pieri type formulas in both ....

I. Schur, Uber die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen, J. reine angew. Math., 139 (1911), pp. 155--250.

.... due to Hiller and Boe [BH] whose proof used the Chevalley formula [C] Another proof, using the Leibnitz formula for symplectic and orthogonal divided differences, was given by Pragacz and Ratajski [PR93] These formulas also arise in the theory of projective representations of symmetric groups [Sc, HH] as product formulas for Schur P and Q functions, and were first proven in this context by Morris [M] The connection of Schur P and Q functions to geometry was established by Pragacz [P88] see also [J] and [P91] In Section 1, we give the basic definitions and state the Pieri type formulas ....

I. Schur, Uber die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen, J. reine angew. Math., 139 (1911), pp. 155--250.

....que est strictement contenue dans l espace lin eaire des fonctions quasi sym etrique gauchis. Puis nous montrons que est une coalg ebre et calculons les dimensions des composantes de degr ee n. 1. Introduction Schur Q functions rst arose in the study of projective representations of S n [8]. Since then they have appeared in variety of contexts including the representations of Lie superalgebras [9] and cohomology classes dual to Schubert cycles in isotropic Grassmanians [4, 7] While studying the duality between skew Schur P and Q functions and their connection to the Schubert ....

I. Schur,  Uber die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen, J. reine angew. Math. , 139 (1911) 155-250.

....none other than the entries of the transition matrix from the basis P (q) to the basis s . Apart from the Hall algebra, the best known applications of Hall Littlewood functions include the character theory of finite linear groups [Gr] projective and modular representations of symmetric groups [S], Mo2] and various topics in algebraic geometry [HS] He] Lu1] Lu2] The aim of these notes is to present some applications to the ordinary representation theory of S n and of GL(n; C) with emphasis on the combinatorial aspects. In particular, the interpretation of Kostka Foulkes polynomials ....

I. Schur, Uber die Darstellung der symmetrischen und der alternierenden Gruppen durch gebrochene lineare Substitutionen, J. reine angew. Math. 139 (1911), 155-250.

....m;n0 Q (m;n) t m 1 t n 2 : Note that Q (m;n) GammaQ (n;m) and Q (n;0) q n (if n 0) Now let = 1 Delta Delta Delta l 0) be a strict partition. If l is odd, set l 1 = 0, and replace l by l 1. Thus in all cases, l is even. The Q function indexed by , as defined by Schur in [S], is given by Q = Pf[Q ( i ; j ) 1i;jl : More generally, the skew Q function Q = can also be defined as the Pfaffian of a matrix whose nonzero entries are Q (m;n) s [JP] For any shifted skew shape = a shifted P 0 tableau T of shape = is (as in Section 2.4) an enriched D 0 = ....

I. Schur, Uber die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen, J. Reine Angew. Math. 139 (1911), 155--250.

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I. Schur, Uber die Darstellung der symmetrischen und der alternirenden Gruppe durch gebrochene lineare Substitutionen, J. Reine Angew. Math. 139 (1911), 155-- 250.

....two families of symmetric functions known as Schur s S functions and P functions. The use of S functions in the character theory of the symmetric groups S n is wellknown [25] and Schur introduced the P functions to play a similar role for the spin characters of the double covering groups of S n [34, 26]. Since that time S functions and P functions have appeared in a number of topics: functions of the KP and BKP hierarchies of soliton equations [13, 40] irreducible characters of the Lie algebras gl n and the Lie superalgebras Q(n) 35] cohomology of grassmannians and isotropic grassmannians ....

I. Schur, Uber die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene linear Substitutionen, J. Reine Angew. Math. 139 (1911), 155-250.

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I. Schur, " Uber die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen," J. reine ang. Math. 39 (1911) 155--250, (Gesammelte Abhandlungen 1, pp. 346--441, Springer-Verlag, Berlin/New York, 1973).

No context found.

I. Schur,  Uber die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen, J. reine angew. Math. , 139 (1911) 155-250.

No context found.

I. Schur,  Uber die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen, J. Reine Angew. Math. 139 (1911), 155{ 250.

No context found.

Schur, I. (1911). Uber die Darstellung der symmetrischen und der alternierenden Gruppen durch gebrochene lineare substitutionen. Journal fur die reine und angewandte Mathematik 139, 155-250. On the representation of the symmetric and alternating group by fractional linear substitutions. Translated by M.-F. Otto. International Journal of Theoretical Physics , this issue.

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