Jordan-Holder Series Research Articles

An Upper Bound for the Lower Central Series Quotients of a Free Associative Algebra

Feigin and Shoikhet conjectured in [4], Feigin and Shoikhet, Mathematical Research Letters, 14,2007,781-95 that successive quotients B m (A n ) of the lower central series filtration of a free associative algebra A n have polynomial growth. In this paper we give a proof of this conjecture, using the structure of a representation of W n , the Lie algebra of polynomial vector fields on , on B m (A n ) which was defined in [4]. Moreover, we show that the number of squares in a Young diagram D corresponding to an irreducible W n -module in the Jordan-Holder series of B m (A n ) is bounded above by the integer , which allows us to confirm the structure of B 3 (A 3 ) conjectured in [4].

Bounds on Weights of Nearby Cycles and Wakimoto Sheaves on Affine Flag Manifolds

We study certain nearby cycles sheaves on an affine flag manifold which arise naturally in the Beilinson–Gaitsgory deformation of the affine flag manifold to the affine Grassmannian. We study the multiplicity functions we introduced in an earlier paper, which encode the data of the Jordan-Holder series. We prove the multiplicity functions are polynomials in q, and we give a sharp bound for their degrees. Our results apply as well to the nearby cycles in the p-adic deformation of Laumon–Haines–Ngo, and also to Wakimoto sheaves.

Inner Derivations of Division Rings and Canonical Jordan form of Triangular Operators

Let D be a division ring and k its center. We show that a generalized canonical Jordan form exists for triangularizable matrices A over D which are algebraic over k, i.e, satisfy f(A) = 0 for some nonzero polynomial f over k. This canonical form is a direct sum of generalized Jordan blocks Jrn (a, /3). This block is an m by m matrix whose diagonal entries are equal to a, those on the first superdiagonal are equal to /3, and all other entries are equal to zero. If a is separable over k then we can choose /3 = 1, but in general this cannot be done. Notation. D denotes a division ring, and k its center. By X we denote a fixed element of D, and by K the centralizer of X in D. We set D* = D {0}, K * = K {0} and U = { Xa aX: a E D }. The right D-vector space whose elements are column n-vectors over D is denoted by D n and its standard basis is denoted as usual by { el, e2, . . ,en }. By R = D[t] we denote the ordinary polynomial ring over D in the indeterminate t. If f E k[t] then (f ) will denote the ideal of k[t] generated by f while fR ( = Rf ) will denote the ideal of R generated by f. Following [7] and [3] we shall say that an R-module is bounded if its annihilator (also called its bound) is not the zero ideal. For a, /8 E D we denote by Dn(a, /3) the vector space Dn considered as a right R-module in which t acts as a D-linear transformation such that elt = ela, and eit = ei a + ei_-/3 for 1 < i < n. Clearly, if /8 # 0 this module is cyclic with generator en. The 1-dimensional module Dl(a, ,8) is independent of /3 and will be denoted by D(a). The module Dn(a, ,8) has length n and each factor of its Jordan-Holder series is isomorphic to D(a). Mn(D) is the k-algebra of n by n matrices over D. X Y for X, Y E Mn(D) means that X is similar to Y. Let Jm(a, 8) E Mm(D) denote the generalized Jordan block with diagonal entries equal to a and those on the first superdiagonal equal to /3, while all other entries are zeros. We shall write Jm((a) for the usual Jordan block Jm(a, 1). For a E D we define [a] to be the set { 8a-y: /3, y E K }. Note that, in general, [a] is not closed under addition. Received by the editors November 16, 1983 and, in revised form, February 22, 1984 and September 10, 19F4. 1YSO Mathematics Subject Classification. Primary 16A42, 16A72; Secondary 12E15, 16A39. Kev words and phrases. Annihilator, Jordan-Holder series, generalized Jordan blocks, irreducible polynomial, bounded module, similarity. 'Research supported by NSERC Grant A-5285. ?1985 American Mathematical Society 0002-9939/85 $1.00 + $.25 per page

On the Kazhdan-Lusztig polynomials for affine Weyl groups

In 191, Kazhdan and Lusztig associated to a Coxeter group W certain polynomials P,,, (y, w E W). In the case where W is a Weyl group, these polynomials (or their values at 1, P,,,,(l)) give multiplicities of irreducible constituents in the Jordan-Holder series of Verma modules for a complex semisimple Lie algebra corresponding to W (see [ 3,4]). Also, in the case where W is an affine Weyl group, it has been conjectured by Lusztig [lo] that these polynomials or their values P,,,(l) enter the character formula of irreducible rational representations for a semisimple algebraic group (corresponding to W) over an algebraically closed field of prime characteristic. The purpose of this paper is to give some formula (4.2, see also 4.10) concerning the Kazhdan-Lusztig polynomials P,,, for affine Weyl groups which appear in this Lusztig conjecture; namely P,,, for y, w “dominant.” (This is a generalization of the q-analogue of Kostant’s weight multiplicity formula in [8].) As a corollary of the formula, we shall show that the Lusztig conjecture above is consistent with the Steinberg tensor product theorem [ 15 1. To be more precise, let P be the weight lattice of a root system R (II) R ‘, a positive root system). The Weyl group W of R acts on P. Hence we can define the semidirect product of W by P, p= W D( P. The element of @ corresponding to A E P is denoted by t,. The group @ contains an afline Weyl group W, = W D( Q as a normal subgroup, where Q is the root lattice. Although # is not a Coxeter group in general, we can define the Bruhat order >, the length function I: p+ La, and the Kazhdan-Lusztig polynomials P,., (y, w E I-V) as natural extensions of those for W,. The main result of this paper (under the specialization q --+ 1) is stated as follows (Corollary 4.10):

On a theorem of Baer, Schreier, and Ulam for permutations

If w = W(Xl ,...) x,) is a word in free variables xi ,..., X, of a group, it leads to complicated (quite often combinatorial or unsolvable) problems to ask for those elements g of a given group G which are expressible in the form g = a?1 ,..., gJ for some g, ,..., g, E G. In the case of commutators w = X, 0 x2 = -1 -1 Xl . x2 . xi x2 we know of finite groups G and g E G’ (= commutator subgroup) such that g f w(g, , g2) for all g, , g, E G; cf. Huppert [12, p. 2581. More general results of this type are to be found in Hall’s lecture notes [ll] or for arbitrary words in Griffith [lo], Rhemtulla [ 151, or Wilson [ 181. This was exploited in Gobel [7, 81. Again, for w = xi o x2 it was shown dually that each element is expressible by commutators in the cases of permutation groups S, , A, (n > 5); cf. Ore [14] and Ito [13]. Analogous results arc known for matrix groups; cf. Clowes and Hirsch [6]. Hence every element of S, is a product of four elements g, 08, from two conjugacy classes gfm and gin. The analogy is true for A, ; cf. HsiiCh’eng-hao [5]. This result was generalized and carried over to the countable case SK0 of all permutations on N by Bertram [3] : (*) Let p be any permutation of SsO with infinite support. Then every permutation of SE0 is a product of four permutations, each conjugate to p. This theorem reflects-in a very strong version-the fact that SX, has only “very few” normal subgroups as known since 1933 from Schreier and Ulam [16]. From Baer’s result [I] we know the Jordan-Holder series in the general case S, of all permutations on a set of cardinality K > K, , which is even unique if this series is finite. Therefore a generalization of (*) for K > X, is to be expected, where, of course, the cardinalities associated with permutations are to be taken in account. Surprizingly, the cardinality 1 s 1 of the support of a permutation scS*, will be sufficient; the support of s is the set of those elements of the underlying set which are not fixpoints of s:

Composition series and intertwining operators for the spherical principal series. II

In this paper, we consider the connected split rank one Lie group of real type F 4 {F_4} which we denote by F 4 1 F_4^1 . We first exhibit F 4 1 F_4^1 as a group of operators on the complexification of A. A. Albert’s exceptional simple Jordan algebra. This enables us to explicitly realize the symmetric space F 4 1 / Spin ( 9 ) F_4^1/{\text {Spin}}(9) as the unit ball in R 16 {{\mathbf {R}}^{16}} with boundary S 15 {S^{15}} . After decomposing the space of spherical harmonics under the action of Spin ( 9 ) {\text {Spin}}(9) , we obtain the matrix of a transvection operator of F 4 1 /Spin ( 9 ) F_4^1{\text {/Spin}}(9) acting on a spherical principal series representation. We are then able to completely determine the Jordan Holder series of any spherical principal series representation of F 4 1 F_4^1 .

Weights of modular representations

Jordan-Holder series of p-representations of Lie algebras of classical type are studied in this note.