G. James and A. Kerber, The representation theory of the symmetric group, Addison-Wesley, 1981.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... xed (linked) block of C 1 : The irreducible characters and the conjugacy classes of Sn are labelled canonically by the partitions of n: If is a partition of n then denotes the irreducible character of Sn ; labelled by : We associate to its ( core and its ( quotient : See [6], Section 2.7. The core is obtained from by removing all hooks from : The number of hooks to be removed from to go to the core is called the ( weight of and denoted w : The quotient is an tuple of partitions ( 0 ; 1 ; 1 ) whose cardinalities add up to w : ....

....are given later. Lemma 3.2. If m 6= 0, then and have the same core. Proof: The lemma follows from the well known fact that the removal of one hook of length r may also be accomplished by removing a sequence of r hooks of length : This fact is seen easily e.g. using the abacus ([6], Section 2.7. See also Theorem (3.3) in [10] When is a partition of v, v n; and ; are partitions of n, we de ne g2S (g) 0 (g) the contribution of the cycle section of type to the inner product of the two irreducible characters. If we take C = S n ; this is in ....

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G. James, A. Kerber, The representation theory of the symmetric group, Addison-Wesley 1981.

....a , b n = b . Proposition 1: We have that n ) an = b n . Proof: See also [6] By column orthogonality for the irreducible characters of Sn , n Xn is a diagonal matrix with the integers z , n in the diagonal. It follows that in the above notation det(Xn ) #n z = an b n . By [2], Corollary 6.5 we n ) an . The result follows. Another proof of the fact that an = b n for all n may be found in [3] We choose an integer # 2, which is fixed from now on. Several concepts below, like regular, singular, defect etc. refer to the integer #. A partition is called regular if ....

G. James, The representation theory of the symmetric groups, Lecture notes in mathematics 682, Springer-Verlag 1978.

....on Fn : f: Pn ) IR is The fact mentioned earlier that r( n) is a reversing measure for the DCF means precisely that K is selfadjoint with respect to this inner product. The following basic facts regarding the character theory of , as well as the full theory, can be found, for example, in [10], and their relevance to random group actions (such as transpositions in our case) in [4] and [6] The characters X of Sn (traces of the irreducible representations) 12 are functions on , constant on conjugacy classes, and as such can be seen to be functions on 7v. They are orthonormal under ....

G. James and A. Kerber, The Representation Theory of the Symmetric Group, AddisonWesley, Reading, Massachusetts (1981).

....(linked) # block of C 1 . The irreducible characters and the conjugacy classes of S n are labelled canonically by the partitions of n. If # is a partition of n then # # denotes the irreducible character of S n , labelled by #. We associate to # its (# )core # # and its (# )quotient # # . See [6], Section 2.7. The # core is obtained from # by removing all # hooks from #. The number of # hooks to be removed from # to go to the core is called the (# )weight of # and denoted w # . The quotient # # is an # tuple of partitions (# 0 , # 1 , # # 1 ) whose cardinalities add up to w # . ....

....details are given later. Lemma 3.2. If m 0, then # and have the same # core. Proof: The lemma follows from the well known fact that the removal of one hook of length #r may also be accomplished by removing a sequence of r hooks of length #. This fact is seen easily e.g. using the # abacus ([6], Section 2.7. See also Theorem (3.3) in [11] When # is a partition of v, v# n, and #, # are partitions of n, we define g#S # # (g)# # # (g) the contribution of the # cycle section of type # to the inner product of the two irreducible characters. If we take n , this is in ....

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G. James, A. Kerber, The representation theory of the symmetric group, Addison-Wesley 1981.

....of partitions n of the integer n and P (q) p(n)q the corresponding generating function. When is a partition of n then l( is the length (number of parts) of . We consider the integers l(n) l( the total length for n. Using the conjugacy map on partitions (see e.g. 1] [5]) l(n) is also the sum of the rst parts of all n. It is convenient to use the exponential notation for partitions. Write = 1 ; 2 ; where a i ( is the multiplicity of i as a part in the partition . Thus l( a i ( The following result is known (see [9] but we ....

....regular if a i ( 0, whenever eji. In that case we write e n. When p is a prime, the p class regular partitions of n give the cycle types of the conjugacy classes of p regular elements in S n . The generating function P e (q) for the number p e (n) of e class regular partitions of n is (see [5]) P e (q) P (q) P (q ) 5) We re ne the de nition of l(n) above to l e (n) l( Then we have Proposition 2.2 Let L e (q) l e (n)q be the generating function for l e (n) L e (q) P e (q)T e (q) Proof. We modify the proof of Proposition 2.1 to see that for an integer ....

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G. James, A. Kerber, The representation theory of the symmetric group, Addison-Wesley 1981.

....S kn . Thus we have the induction homomorphism fi = Ind : R( Sn ]S k ) R(S kn ) Definition. The outer plethysm on R(S) is a map [M ] R(S) R(S) given by [M ] N ] Ind (M Omega k Omega N ) where M is an Sn module and N is an S k module. In James Kerber s notation [3], the outer plethysm [M ] N ] is denoted by [M ] fi [N ] In [8] this outer plethysm is used to construct operations on R(S) and it is shown that with respect to these operations R(S) is a special ring (see [1, 2, 4] for definitions and basic results about rings) The operations : ....

G. James and A. Kerber, "The Representation Theory of the Symmetric Group," Encyclopedia of Mathematics and its Applications, No. 16, Reading, Massachusetts, Addison Wesley, 1981.

....RING OF THE GENERAL LINEAR GROUP Essam A. Abotteen Emporia State University The investigation of plethysms (inner and outer) in the representation theory of finite classical groups has been one of the important outstanding problems in the representation theory of the symmetric group [5], 7] and [8] The fundamental theorem of the representation theory of the symmetric group has been more or less known since the origins of the subject with Frobenius at the turn of the century. This theorem states there is an isomorphism between the representation ring of the symmetric groups Sn ....

G. James and A. Kerber, The Representation Theory of the Symmetric Group, Addison--Wesley, New York, 1981.

....is a new combinatorial result (Proposition (4.5) about the connection between certain p invariants of a partition. We also need a combinatorial description of the power of p dividing # # (1) essentially due to Macdonald [9] Remark. It is an easy consequence of the Murnaghan Nakayama formula ([6], 2.4.7) that any non linear character of S n vanishes on either an n cycle or on an (n 1) cycle and thus the corresponding conjugacy classes are strongly orthogonal as defined in the next section. Most likely it is also true, that any nonlinear character of A n vanishes on some element of ....

....that any nonlinear character of A n vanishes on some element of prime order, but the method used here is not su#cient to prove this. For the following calculations p need not be a prime. To each partition # of n is associated a p core C p (#) and a p quotient Q p (#) # 0 , # p 1 ) See [6], 14] The p core is a partition without p hooks obtained by removing a sequence of p hooks from #. The p quotient is a p tuple of partitions. We may recover # from C p (#) and Q p (#) The following important fact is needed: 4.3) Lemma. Let H p (#) be the (multi )set of hooks of # of length ....

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G. James, A. Kerber, The representation theory of the symmetric group, Addison-Wesley Publishing Co., Reading, Mass., 1981.

....a (C G; S r ) bimodule. The irreducible characters of S r are parametrized by partitions of r, and we denote by [ the character corresponding to the partition . For each such partition we consider the [ isotypic part in and denote it by V [ Its character is given by the formula (see [19], 5.2.13) g) r 2Sr [ g a i ( for g 2 G; where a i ( denotes the number of cycles of length i in the permutation 2 S r . For each r 0 and each partition of r, we have thus defined an operation, 7 [ on the set of actual characters of G. We can use the ....

....of a finite irreducible Coxeter group can be modified by a group automorphism so that it fixes the reflection character ae. It then remains to use the known results on Aut(G) see [17] Satz II.5. 5, for example) The other, more direct approach is based on the following observation (see [19], Theorem 2.4.10) a) For n 6= 1; 2; 3; 6 there are only two irreducible characters of degree n Gamma 1: these are the characters ae and sgn. b) The group S 6 has four irreducible characters of degree 5. Now let 2 AutCT(G) be arbitrary. We want to show that is trivial if n 6= 6. We ....

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G. D. James and A. Kerber, The Representation Theory of the Symmetric Group, Encyclopedia of Math. 16, Addison--Wesley, 1981.

.... map which is the fundamental isometry between the center of the group algebra of 5 , C(5 , and the space of homogeneous symmetric functions of degree n, see [G R] or [Macdonaldl] Also 52( is the character of the irreducible polynomial representation of GL, C) corresponding to , see [J K]. There are two standard ways to define the Schur function 2( for = 0) a partition of m n. The classical definition is to define 52( as a quotient of alternates. der x n j l i,j n = der x J li,jn The second definition is the combinatorial definition which is ....

G. James and A. Kerber, "The Representation Theory of the Symmetric Group", Addison-Wesley Pub. Corn., Reading, Mass, 1981.

....algebra H(S (s) of type A. Because q this completes the proof when t = t . If t B t then there exists an integer i, 1 i n, such that t = t(i; i 1) B t; then st = S st 0 T i and the result follows by the argument of [6, Theorem 3. 15] We next generalize James notion [11] of ladders to the case of multipartitions. A ladder is an equivalence class of nodes where two nodes (j; k) s and (l; m) t belong to the same ladder k (e Gamma 1)j e s = m (e Gamma 1)l e t ; where we abuse notation and consider e s and e t as integers. In particular, two nodes belong to ....

G. D. James and A. Kerber, The representation theory of the symmetric group, 16, Encyclopedia of Mathematics, Addison--Wesley, Massachusetts, 1981.

....1 ; p 2 ; D f = f( n p n ; In particular we have D s (s ) s = We denote by 2 the ring homomorphism of Sym which sends p i to p 2i . 2 (f) is the plethysm of p 2 by f introduced by Littlewood. The 2 core (2) of a partition is defined as follows (see [2]) If has no 2 hooks = 2) otherwise remove as many 2 hooks as possible from the diagram of ; the partition obtained in this way is (2) We recall that (2) is a staircase partition ae k = k Gamma 1; 1; 0) We can also compute the 2 core using the algorithm that computes the ....

....ffl 2 ( sign(oe) Finally , subtract from the even parts of the corresponding residues in fl and divide by 2 to obtain . The same procedure applied to the odd parts gives the second partition . There exists a one to one correspondence between a partition and its 2 core and 2 quotient [2]. We write = 2) We denote by Sym ; the subspace spanned by Schur S functions indexed by partition without 2 core. We can now define J( and J( At last we introduce the linear involution of Sym ; defined by (s ) Gamma1) ffl 2 ....

G. D. James, A. Kerber, The representation theory of the symmetrics groups,(1981) Addison-Wesley, Readings, Massachusetts.

....1; 1; 1; 1) and charge j 2 ( mod 4) Another combinatorial formulation of Theorems 3.1 and 3.2 can be presented by means of the notion of ribbon tableau, which will also provide the clue for their generalization. 4 Ribbon tableaux To a partition is associated a k core (k) and a k quotient [14]. The k core is the unique partition obtained by successively removing k ribbons (or skew hooks) from . The different possible ways of doing so can be distinguished from one another by labelling 1 the last ribbon removed, 2 the penultimate, and so on. Thus Figure 1 shows two different ways of ....

G. D. James and A. Kerber, The representation theory of the symmetric group, Addison-Wesley, 1981.

....representations The aim of this section and the next is to give a complete description of the role played by the plethysm operation in the representation theory of the symmetric group. For a broader discussion of wreath products (involving more general groups) and their representations, see [JK]. Let m and n be positive integers, and let be the partition in Pi mn consisting of m blocks each of size n; assume the kth block consists of the integers f(k Gamma 1)n 1; kng: The subgroup of Smn which fixes is the wreath product of the group Sm with Sn ; it consists of those ....

....(indexed by partitions of mn) The problem of computing these coefficients is a difficult one; no combinatorial rule is known for the coefficients in general. There is a large literature on solutions for special cases (see the references in [L] and also [CT] In addition to the references in [JK], more recent work includes papers by Christophe Carr e [C] whose thesis is devoted to plethysm, and Carr e and Jean Yves Thibon ( CT] We record some facts about plethysm which follow easily from the definition. Proposition 4.3. 1) pn [p m ] pmn = pm [p n ] 2) p [p m ] pm Delta for ....

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G. D. James and A. Kerber, The Representation Theory of the Symmetric Group, Encyclopedia of Mathematics, vol. 16, Addison-Wesley, Reading, MA, 1981.

.... and W is a G module, we denote by V [W ] the corresponding module for Sn [G] As a vector space V [W ] is simply the tensor product of the nth tensor power of W with V: The action of G is just the diagonal action on the n fold tensor product : For a precise description of the Sn action, see [JK]. In this section we will be concerned only with the case in which V is a one dimensional representation of Sn ; that is, either the trivial representation or the sign representation. Thus V [W ] is isomorphic to as a vector space. An element oe in Sn acts on by permuting the tensor ....

G. D. James and A. Kerber, The Representation Theory of the Symmetric Group, Encyclopedia of Mathematics, vol. 16, Addison-Wesley, Reading, MA, 1981.

....over a p adic field relative to the Iwahori subgroup. i Introduction The classical representation theory of the symmetric group S is well known. The irreducible representations of S over the complex field C are labelled by partitions w of the integer n and realNed through several constructions [6]. One of the most useful realNations employs primitive idempotents in the group ring C. S now known s Young symmetrizers [20] These idempotents in .S correspond to the Young standard tbleux with n entries [11] Suppose that tbleu hs shape . Let S nd T be the subgroups in S respectively preserving ....

G. D. JAMES and A. KERBER, The Representation Theory of the Symmetric Group (Addison-Wesley, Reading, Massachusetts, 1981).

.... work [DR] we shall introduce in this paper Specht modules for an Ariki Koike algebra as submodules of those permutation modules and investigate their basic properties such as Standard Basis Theorem and the ordinary Branching Theorem, generalizing several classical constructions given in [JK] and [DJ] for type A. The second part of the paper moves on looking for modular branching rules for Specht and irreducible modules over an Ariki Koike algebra. These rules for symmetric groups were recently established by Kleshchev [K] and were generalized to Hecke algebras of type A by Brundan ....

....L k = T j,k 1 L k T k 1,r = T j,k T k L k T k 1,r = qT j,k L k 1T 1 k T k 1,r = qL k 1T j,k T 1 k T k 1,r . Now the result (a) follows easily since qT 1 k = T k 1) The statements (b) and (c) have been proved in [DR, 3. 1) The Branching Theorem for symmetric groups can be found in [JK]. The following is the q version (see [Jo, 3.4] 3.5) Lemma. For any # (r) letn 1 , n k be the removable nodes of counted from top to bottom, and define M 0 =0and M t = z # T j n t (S r 1 ) M t 1 for t # 1. Then we have a filtration of (S r 1 ) submodules for S R = z ....

G. James and A. Kerber, The representation theory of the symmetric group, AddisonWesley, London, 1981.

....are embedded in a1 ) acts by p 2k 1 (multiplication by p 2k 1 ) and = p 2k 1 . Thus one obtains in this way the principal realization ae P of the basic representation of b sl 2 , first constructed by Lepowsky and Wilson [23] Using the combinatorics of 2 cores and 2 quotients of partitions [11], which is known to be strongly connected with the basic representation of b sl 2 [1, 18] we introduce a linear involution of Sym such that s ae k 2 (Sym) 9) where ae k denotes the staircase partition (k; k Gamma 1; 1) and 2 is the ring homomorphism of Sym sending p j to p ....

....2 . This will occupy us for the rest of the paper. 7 Partitions and symmetric functions In this section, we collect a number of definitions and propositions relative to the combinatorics of partitions and the algebra of symmetric functions. Most of these are standard (our main references being [11, 25, 30]) or elementary. This background will be useful in the following sections when we get to the description of the basic representation of b sl 2 and its connection with the operators Delta k . 18 7.1 Partitions and sequences Let be a partition and k be a fixed integer. We associate with and k ....

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G. James, A. Kerber, The representation theory of the symmetric group, Addison Wesley, 1981.

....is also considerable knowledge on some series of simple groups. For the reason of completeness we brie y sketch the known results. Various authors, among them Robinson, Kerber and James have established a theory of modular representations for the symmetric and alternating groups (see for example [75, 84]) Burkhardt did some work for groups of Lie type; in particular he found all decomposition numbers for the simple groups PSL(2; q) 15] all decomposition numbers for the Suzuki groups in odd characteristics [16] and some decomposition numbers for the Suzuki groups and the unitary groups U(3; ....

G. D. James, The Representation Theory of the Symmetric Groups, Lecture Notes in Mathematics vol. 682, Springer-Verlag, Berlin, 1978.

....zwischen Tableaux Die Operation ps i gibt eine Bijektion zwischen Tableaux vom Umri I und Gewicht w 1 ; w i ; w i 1 ; wn und den Tableaux vom gleichen Umri und dem Gewicht w 1 ; w i 1 ; w i ; w i 2 ; wn . Man vergleiche diese Bijektion mit der Bijektion in James Kerber [JK81] S.90, es werden die gleichen Tableaux gepaart. So wird z.B. das Tableau 35 245 1233 ffi ffiffi ffi ffiffi mittels ps 1 nach ffi ffiffi ffi ffi fflffi 35 245 1133 abgebildet, oder z.B. mittels ps 2 nach ffi ffi ffi fflffiffi 35 245 1223: Dabei wurden die verschobenen Steine besonders ....

Gordon James & Adalbert Kerber, The Representation Theory of the Symmetric Group, Addison-Wesley, Reading Massachusetts, 1981

.... natural central element c m 2 Hm (i) 3] Indeed, if M is a (i) module, the restricted module M # Hm Gamma1 (i) splits into a direct sum of eigenspaces of c m Gamma1 , which we denote by M # k ; 0 k n) These k restriction operators were first defined for symmetric groups by Robinson (see [14]) More generally, we write M # k for the Hm Gammaj (i) module obtained from M by j successive k restrictions. Let D be a simple Hm (i) module, and let j = j 0 ; j n Gamma1 ) be another n tuple of nonnegative integers. We say that D satisfies the generalised Jantzen Seitz condition ....

G.D. James and A. Kerber, The representation theory of the symmetric group, AddisonWesley, 1981.

.... observing that the determinant of the character table of W n is a non zero integer, hence remains non zero under any specialization to a field of characteristic 0, and that the cardinality of P (n 1) e equals the number of e regular partitions of n 1 (note that the proof of this fact in [8], Lemma 6.1.2, works also for e instead of a prime p) Note that the same arguments also work for specializations into a finite field of characteristic r (inducing the r modular decomposition map in the sense of [6] where r is a prime not dividing the order of W n . One is tempted to call an ....

G. D. James and A. Kerber, "The Representation Theory of the Symmetric Group", Encyclopedia of Math. 16, Addison--Wesley, 1981.

....of the identity into the sum of primitive central idempotents, the group algebra kS n decomposes into the direct sum of blocks, and the category of kS n modules decomposes into the direct sum of the module categories of these blocks. By the general theory on Specht modules S , it is well known [4, 10] that (1) Reduction modulo r of any OS n submodule of an irreducible kS n module S (jj = n) has the same set of composition factors all belonging to a single block. 2) Blocks are parametrized by r cores and S belongs to the block corresponding to the r core c of . 14 (3) The ....

G. James and A. Kerber, 'The Representation Theory of the Symmetric Groups' Addison-Wesley (1981)

....in a hyperplane given by an equation of the form y i Gamma y j = ml for some i; j with 1 i; j k and with m 2 Z. We next mention the relation between the orbits of the W k action and the notion of the l core of a diagram. We recall the notions of a rim hook and l core, for example from [JK]. For (a; b) a node of a Young diagram , the corresponding hook is the portion of row a to the right of (a; b) together with the portion of column b below (a; b) including the cell (a; b) the length of the hook is h (a;b) a Gamma a b Gamma b 1. The corresponding rim hook is the ....

....l. 10 Let be a diagram with no more than k rows. The point y = ae 2 D is determined by its set of components fy i g. It is not difficult to see that the operation of removing a rim hook of length l from corresponds exactly to the operation of reducing one element of fy i g by l, see [JK], Lemma 2.7.13. The following result must be well known. 2.4.1 Lemma. Let and be diagrams of the same size with no more than k rows. Let y = ae, z = ae. The following are equivalent: 1) and have the same l core. 2) For 0 r l Gamma 1, jfi : y i j r mod lgj = jfi : z i j r mod lgj. ....

G. James and A. Kerber, The Representation Theory of the Symmetric Group, Addison-Wesley, 1981.

....; 25) then follows in the = 0 case, as a direct corollary: Pr ch V ( 0 ) j1 j6j0 modn 1 Gamma q Proof: By Theorem 2.10 and Definition 2.8.2, the coefficient of q on the right side of (25) is given by the number of n regular partitions of N . The generating function for this number [25] then gives the result. We will later describe a realisation of V ( 0 ) that has a basis naturally indexed by the set of n regular partitions Pi n . 2.4 Restricted paths In Definition 2.2, we defined a set of unrestricted paths P ( 0 ) We now use these to define restricted paths. Definition ....

....0, the inequivalent irreducible representations of Sm are indexed by Pi(m) the set of partitions of m. There exist a number of ways to calculate the matrices of the irreducible representation of Sm corresponding to a particular 2 Pi(m) One way is by means of the Specht module S (see [25] for example) whereby the resulting matrices contain only integers. On restricting to Sm Gamma1 , the module S is no longer irreducible in general. One has the following well known branching rule. The definition of q binomial coefficient here is different from the one used in x4. 23 ....

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G.D. James and A. Kerber, The representation theory of the symmetric group, Addison-Wesley, 1981.

....[7] Read [9] and Hughes [2] respectively. However, there are well known constructions of irreducible representations and of irreducible modules, called Specht modules, for the symmetric groups S n which are based on elegant combinatorial concepts connected with Young tableaux, etc. see, e.g. [3]) Therefore, in this paper we show how the Specht approach to the irreducible representations of the symmetric groups can be extended to deal with the generalised symmetric groups G(m; 1; n) As a convention, throughout this paper, we assume that is a primitive m th root of unity. 1. Diagrams, ....

James, G. D.: The Representation Theory of the Symmetric Groups. Lecture Notes in Mathematics, vol. 682, Springer-Verlag, New York 1978.

....is acting as a permutation group on the basis of the space in If you want to check this example, then please put this input into a file input, say, and enter (after having the test program compiled) a. out input output You will then find at the end of the file output the following lines: [1,3] [ 3: 1: 6:2:2:2: 6:2:2: 6: 6:2: 6:2: 6: 6:2:2: The first of these lines says that the permutation representation of S4 decomposes into two irreducible representations D and D2 (according to the numbering in the input file) that their degrees are 1 and 3, respectively, and ....

....that the permutation representation of S4 decomposes into two irreducible representations D and D2 (according to the numbering in the input file) that their degrees are 1 and 3, respectively, and so the transformed operator decomposes into 4 blocks, three of which are equal. Thus the first row [1, 3] shows the multiplicities of the blocks in the transformed operator. Note that, by theory, these multiplicities are equal to the dimensions of the irreduciblesl Finally the blocks are given, each of them just once. Therefore, in our case, the transformed operator looks as follows: 3 0 0 0 0 1 0 ....

G.D. James and A. Kerber, The Representation Theory of the Symmetric Group (Reading Massachusetts, 1981)

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G. James and A. Kerber, The representation theory of the symmetric group, Addison-Wesley, 1981.

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G. James, The representation theory of the symmetric groups, Lecture notes in mathematics 682, Springer-Verlag 1978.

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G. James and A. Kerber, The Representation Theory of the Symmetric Group, Addison-Wesley, New York, 1981.

No context found.

G. James and A. Kerber. The Representation Theory of the Symmetric Group. Addison-Wesley, Reading, MA, 1981.

No context found.

G. James, The representation theory of the symmetric groups, Lecture notes in mathematics 682, Springer-Verlag 1978.

No context found.

G. James and A. Kerber, \The Representation Theory of the Symmetric Group," Addison-Wesley, Reading, MA, 1981.

No context found.

G. D. James and A. Kerber, The Representation Theory of the Symmetric Group. Encyclopedia of Math. and its Appl. Vol. 16, AddisonWesley, 1981.

No context found.

G. James and A. Kerber, The Representation Theory of the Symmetric Groups. Addison-Wesley, London, 1980.

No context found.

G. D. James, The representation theory of the symmetric groups, Lecture Notes in Math., vol. 682, Springer-Verlag, 1978.

No context found.

G. James and A. Kerber, The Representation Theory of the Symmetric Group, Addison{ Wesley, Reading, Massachusetts (1981).

No context found.

G. James and A. Kerber (1981), The Representation Theory of the Symmetric Group, Addison-Wesley, Reading, Massachusetts.

No context found.

G. D. James. The Representation Theory of the Symmetric Groups, volume 682 of Lecture Notes in Mathematics. Springer-Verlag, 1978.

No context found.

G. D. James and A. Kerber, The Representation Theory of the Symmetric Group. Encyclopedia of Math. and its Appl. 16, AddisonWesley, 1981.

No context found.

G. James, The representation theory of the symmetric groups, Lecture notes in mathematics 682, Springer-Verlag 1978.

No context found.

G. James, The representation theory of the symmetric groups, Lecture notes in mathematics 682, Springer-Verlag 1978.

No context found.

G. D. James and A. Kerber, The Representation Theory of the Symmetric Group, (Addison-Wesley, Reading, 1981).

No context found.

G. James and A. Kerber, The Representation Theory of the Symmetric Group, Addison-Wesley, 1981.

No context found.

G.D. James and A. Kerber, The Representation Theory of the Symmetric Group, (Encyc. of Math. and its Applications, Addison-Wesley, 1981).

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G. D. James and A. Kerber, The Representation Theory of the Symmetric Group, Encyclopedia Math. 16 (1981).

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G. D. James and A. Kerber, The representation theory of the symmetric group, 16, Encyclopedia of Mathematics, Addison--Wesley, Massachusetts, 1981.

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G. D. James and A. Kerber, The representation theory of the symmetric group, 16, Encyclopedia of Mathematics, Addison--Wesley, Massachusetts, 1981.

No context found.

G. D. James, The representation theory of the symmetric groups, SLN, 682, Springer--Verlag, New York, 1978.

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G.D. James, The representation theory of the symmetric groups, Lecture Notes in Mathematics 682, Springer, 1978.

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