Showing papers on "Symmetric group published in 1985"

Random Shuffles and Group Representations

Abstract: This paper considers random walks on a finite group $G$, in which the probability of going from $x$ to $yx, x, y \in G$, depends only on $y$. The main results concern the distribution of the number of steps it takes to reach a particular element of $G$ if one starts with the uniform distribution on $G$. These results answer some random sorting questions. They are attained by applications of group representation theory.

Classification of 2-transitive symmetric designs

Abstract: All symmetric designs are determined for which the automorphism group is 2-transitive on the set of points.

On the decomposition numbers of the finite general linear groups. II

Abstract: Let G = GL,,(q), q a prime power, and let r be an odd prime not dividing q. Let s be a semisimple element of G of order prime to r and assume that r divides qdeg(X\) 1 for all elementary divisors A of s. Relating representations of certain Hecke algebras over symmetric groups with those of G, we derive a full classification of all modular irreducible modules in the r-block Bh of G with semisimple part s. The decomposition matrix D of Bs may be partly described in terms of the decomposition matrices of the symmetric groups corresponding to the Hecke algebras above. Moreover D is lower unitriangular. This applies in particular to all r-blocks of G if r divides q 1. Thus, in this case, the r-decomposition matrix of G is lower unitriangular. Introduction. The modular representation theory of finite groups of Lie type defined over a field of characteristic p is naturally divided into the cases of equal characteristic r = p and unequal characteristic r / p. This paper begins the study of decomposition numbers in the unequal characteristic case of the finite general linear groups when r > 2. Let G be the full linear group of degree n over GF(q) and 2 /= r be a prime not dividing q. Let s c G be semisimple and consider the geometric conjugacy class (s)G of irreducible characters corresponding to the G-conjugacy class of s. Using basic properties of the Deligne-Lusztig operators introduced in [7] we will determine a parabolic subgroup P = Ps of G and a cuspidal irreducible character of X of P such that (s)G is just the set of constituents of XG, the induced character. Let { R, R, K } be an r-modular system for G and M a KP-module affording X. Then End KG(M ) is called the Hecke algebra of M (compare [5]) and has been calculated in a more general context by R. B. Howlett and G. I. Lehrer in [12]. It turns out that End KG(M )_ K[W], the Hecke algebra of W, where W = WJ denotes the Weyl group of CG(s) (the centralizer of s in G) (compare [1, 2, 4, 13]). In particular, K[W] has a basis { Tw, Iw c W } such that the structure constants of the multiplication are all contained in R c K. So let R[W] be the R-order in K[W] generated by wTIw EW} and R[W] = R ?R R[W]. Choosing a special RP-lattice S in M, and using the classification of the r-blocks of RG given by P. Fong and B. Srinivasan in Received by the editors July 25, 1983 and, in revised form, October 18, 1983, May 23, 1984 and September 1, 1984. 1980 Mathematics Subject Classification. Primary 20C20; Secondary 20G40. 'This work was supported by the Deutsche Forschungsgemeinschaft. ?1985 American Mathematical Society 0002-9947/85 $1.00 + $.25 per page

A note on decomposition numbers for general linear groups and symmetric groups

Abstract: In [5] James proved theorems on the decomposition numbers, for the general linear groups and symmetric groups, involving the removal of the first row or column from partitions. In [1] we gave different proofs of these theorems based on a result valid for the decomposition numbers of any reductive group. (I am grateful to J. C. Jantzen for pointing out that the Theorem in [1] may also be derived from the universal Chevalley group case, which follows from the proof of 1 ·18 Satz of [6] – the analogue of equation (1) of the proof being obtained by means of the natural isometry (with respect to contra-variant forms) between a certain sum of weight spaces of a Weyl module V (λ) of highest weight λ and the Weyl module corresponding to λ for the Chevalley group determined by the subset of the base involved.) However, we have recently noticed that this result for reductive groups, even when specialized to the case of GL n , gives a substantial generalization of James's Theorems. This generalization, which we give here, is an expression for the decomposition number [λ: μ] for a pair of partitions λ, μ whose diagrams can be simultaneously cut by a horizontal (or vertical) line so as to leave the same number of nodes above the line (or to the left of the line for a vertical cut) in both cases. Cutting between the first and second rows gives James's principal of row removal ([5], theorem 1) and cutting between the first and second column gives his principle of column removal ([5], theorem 2). Another special case of our horizontal result, involving the removal of bottom rows of a pair of partitions, is stated in [7], Satz 8.

Normal subgroups of doubly transitive automorphism groups of chains

Abstract: We characterize the structure of the normal subgroup lattice of 2-transitive automorphism groups A(Q) of infinite chains (Q, <) by the structure of the Dedekind completion (Q, <) of the chain (Q7 <). As a consequence we obtain various group-theoretical results on the normal subgroups of A(Q), including that any proper subnormal subgroup of A(Q) is indeed normal and contained in a maximal proper normal subgroup of A(Q), and that A(Q) has precisely 5 normal subgroups if and only if the coterminality of the chain (Q, < ) is countable.

On the complexity of immanants

Abstract: Let χ be an irreducible character of the symmetric group Sn . The immanant corresponding to χis defined by ifχ is the alternating resp. principal character, we get the determinant resp. permanent. While there exist fast algorithms for the computation of the determinant, the permanent seems to be computationally intractable. We investigate the difference between the complexity of the determinant and the permanent by giving lower and upper bounds for the complexity of arbitrary immanants.

The K -functor (Grothendieck group) of the infinite symmetric group

Abstract: The Grothendieck group K0(δ∞) of the group of finite permutations of a countable set is described. All semifinite characters of δ∞ are described and with their help the cone of representations K + 0 (δ∞) is characterized.

On the number of conjugacy classes in finite groups of lie type

Abstract: (1985). On the number of conjugacy classes in finite groups of lie type. Communications in Algebra: Vol. 13, No. 5, pp. 1019-1045.

Generalized dirac identities and explicit relations between the permutational symmetry and the spin operators for systems of identical particles

Abstract: The well known one-to-one correspondence between the eigenstates of the total spin for a system of spin-½ particles and irreducible representations of the symmetric group with up to two rows in the Young shape is the basis of interesting formal developments in quantum chemistry and in the theory of magnetism. As an explicit manifestation of this correspondence the class operators of the symmetric group are demonstrated to be expressible in terms of the total spin operator. This correspondence does not hold for higher elementary spins. The extension to arbitrary spin is investigated using Schrodinger's generalization of the Dirac identity, which expresses the transpositions in terms of two-particle spin operators. It is shown that additional operators, which for σ = ½ reduce to the total spin operator, are needed for a complete classification. Some aspects of the formalism are developed in detail for σ = 1. In this case a classification identical with that provided by the irreducible representations of the symmetric group is obtained in terms of the eigenstates of two commuting operators, one of which is the total spin operator.

Homology of integral upper-triangular matrices

Abstract: We calculate the homology of the multiplicative group of integral upper-triangular n x n matrices at all primes p > n 1. 1. Group homology. For any ring A, let Gln(A) be the general linear group of invertible n x n matrices over A, and Un(A) the subgroup of upper-triangular matrices with ones on the diagonal. The purpose of this note is to prove Theorem 1.1, which gives a calculation of the Eilenberg-Mac Lane group homology of Un(Z) at all primes p > n 1. (This group homology is the same as the homology of the compact nilmanifold Un(R)/Un(Z).) For any permutation a in the symmetric group Sn, let Ma E Gl n(A) denote the matrix with the property that Ma bi = ba(i), where b, is the column vector with one in the ith coordinate and zero in the others. Let Un-(A) be the group of lower-triangular matrices in Gl n(A) with ones on the diagonal, and Un? the intersection (M, 1Un-(A)M,) n Un(A). It is easy to see that Unb(A) is the subgroup of Un(A) given by matrices which have zero in row i, columnj whenever i u(j)} We will denote this cardinality by 1(a). Note that the permutation a is uniquely determined by the set L( a). Let Z(p) stand for the localization of the ring Z at the prime p. 1.1 THEOREM. If p > n 1, then H*(Un(Z),Z(p)) is the free Z(p)-module on the classes Ca, a E Sn. The proof of Theorem 1.1 is in two steps. Let Un denote Un(Z). 1.2 PROPOSITION. The cycles Ca, a E S, generate a free Z-summand of U*(Un, Z) of rank n!. Received by the editors April 17, 1984. 1980 Mathematics Subject Classification. Primary 55R35; Secondary 55S30, 17B56, 20H25. ?1985 American Mathematical Society 0002-9939/85 $1.00 + $.25 per page

An inequality for the second immanant

Abstract: Let be the irreducible complex character of the nth symmetric group corresponding to the partition be the immanant defined on n by n complex matricesby matrice by In this paper we show that if A is positive semidefinite, then where per (A)is the permanent of A.

Examples of noncommutative groups with nontrivial exit-boundary

Abstract: A number of new counterexamples are given, disproving certain assumptions about the mutual relations of the exit-boundary (Poisson boundary) of a random walk on a group and the amenability and growth of the group. Random walks are constructed with nontrivial exit-boundary on the affine group of the dyadic-rational line and on the infinite symmetric group.

Varieties of automorphism groups of orders

Abstract: The group A(ST) of automorphisms of a totally ordered set fi must generate either the variety of all groups or the solvable variety of class n. In the former case, A(fi) contains a free group of rank 2N° ; in the latter case, A(ST) contains a free solvable group of class n — 1 and rank 2K°.

Cubes of conjugacy classes covering the infinite symmetric group

Abstract: Using combinatorial methods, we prove the following theorem on the group S of all permutations of a countably-infmite set: Whenever p 6 S has infinite support without being a fixed-point-free involution, then any s G S is a product of three conjugates of p. Furthermore, we present uncountably many new conjugacy classes C of S satisfying that any s G S is a product of two elements of C. Similar results are shown for permutations of uncountable sets.

Racah-Wigner approach to standardization of permutation representations for finite groups

Abstract: The problem of equivalence of permutation representations of the finite groups is discussed in terms of transforms by bijections of the carrier sets and by group automorphisms. A formal description...