scholarly journals Matrix units associated with the split basis of a Leonard pair

2006 ◽  
Vol 418 (2-3) ◽  
pp. 775-787 ◽  
Author(s):  
Kazumasa Nomura ◽  
Paul Terwilliger
Keyword(s):  
Leonard Pair ◽  
Matrix Units

Ultrastructure of Gene Transcription in Spermatocytes of Trichosia pubescens Morgante, 1969 (Diptera: Sciaridae)

1976 ◽  
Vol 31 (3-4) ◽  
pp. 186-189 ◽  
Author(s):  
J.M. Amabis ◽  
K. K. Nair
Keyword(s):  
Gene Transcription ◽  
Gradual Increase ◽  
Christmas Tree ◽  
Ribosomal Cistrons ◽  
Matrix Units

Abstract In spread preparations of whole spermatocytes of Trichosia pubescens the ribosomal transcription units could be identified by virtue of their “Christmas-tree-like” morphology. The ultrastructural features of these matrices are similar to those described in other eucaryotes. However, in contrast to the previously described systems the “spacer” unit between these matrix units is very small or non-existent at all. In addition, axial fibers displaying much longer lateral fibrils, irregularly spaced, and not as closely packed as in the ribosomal cistrons, were found. These are likely to represent active nonribosomal transcription. In a few instances these lateral fibrils show a gradual increase in their length.

MATRIX UNITS FOR THE GROUP ALGEBRA kGf = k((ℤ2 × ℤ2) ≀ Sf)

2009 ◽  
Vol 02 (02) ◽  
pp. 255-277
Author(s):  
B. Sivakumar
Keyword(s):  
Group Algebra ◽  
Irreducible Representations ◽  
Complete Set ◽  
Matrix Units

The irreducible representations of the group Gf := (ℤ2 × ℤ2) ≀ Sf are indexed by 4-partitions of f, i.e., by the set {[α]3[β]2[γ]1[δ]0|α ⊢ u3, β ⊢ u2, γ ⊢ u1, δ ⊢ u0, u0 + u1 + u2 + u3 = f}. This set is in 1 - 1 correspondence with partitions of 4f whose 4-core is empty. In this paper we construct the inequivalent irreducible representations of Gf. We also compute a complete set of seminormal matrix units for the group algebra kGf.

scholarly journals A characterization of the semigroup of matrix units

1980 ◽  
Vol 29 (3) ◽  
pp. 291-296
Author(s):  
Bernard R. Gelbaum ◽  
Stephen Schanuel
Keyword(s):  
Subject Classification ◽  
M 10 ◽  
20 M 10 ◽  
The Matrix ◽  
Matrix Units

AbstractLet I be a set and let (I) denote the set consisting of the 0 matrix over I × I and the matrix units over I × I. Then for x, z in (I) and x≠0≠z, xyz≠0 has precisely one solution y. This and several other statements are shown to be equivalent characterizations of (I) regarded as a semigroup with zero.1980 Mathematics subject classification (Amer. Math. Soc.): 20 M 10.

scholarly journals The switching element for a Leonard pair

2008 ◽  
Vol 428 (4) ◽  
pp. 1083-1108 ◽  
Author(s):  
Kazumasa Nomura ◽  
Paul Terwilliger
Keyword(s):  
Switching Element ◽  
Leonard Pair

scholarly journals Self-dual Leonard pairs

2019 ◽  
Vol 7 (1) ◽  
pp. 1-19
Author(s):  
Kazumasa Nomura ◽  
Paul Terwilliger
Keyword(s):  
Vector Space ◽  
The Self ◽  
The Other ◽  
Endomorphism Algebra ◽  
Comprehensive Description ◽  
Leonard Pairs ◽  
Positive Dimension ◽  
Linear Maps ◽  
Leonard Pair ◽  
Dual Case

Abstract Let F denote a field and let V denote a vector space over F with finite positive dimension. Consider a pair A, A* of diagonalizable F-linear maps on V, each of which acts on an eigenbasis for the other one in an irreducible tridiagonal fashion. Such a pair is called a Leonard pair. We consider the self-dual case in which there exists an automorphism of the endomorphism algebra of V that swaps A and A*. Such an automorphism is unique, and called the duality A ↔ A*. In the present paper we give a comprehensive description of this duality. In particular,we display an invertible F-linearmap T on V such that the map X → TXT−1is the duality A ↔ A*. We express T as a polynomial in A and A*. We describe how T acts on 4 flags, 12 decompositions, and 24 bases for V.

scholarly journals LEONARD PAIRS AND THE ASKEY–WILSON RELATIONS

2004 ◽  
Vol 03 (04) ◽  
pp. 411-426 ◽  
Author(s):  
PAUL TERWILLIGER ◽  
RAIMUNDAS VIDUNAS
Keyword(s):  
Vector Space ◽  
Linear Transformations ◽  
Leonard Pairs ◽  
Positive Dimension ◽  
The Matrix ◽  
Leonard Pair

Let K denote a field and let V denote a vector space over K with finite positive dimension. We consider an ordered pair of linear transformations A:V→V and A*:V→V which satisfy the following two properties: (i) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A* is diagonal. (ii) There exists a basis for V with respect to which the matrix representing A* is irreducible tridiagonal and the matrix representing A is diagonal. We call such a pair a Leonard pair on V. Referring to the above Leonard pair, we show there exists a sequence of scalars β,γ,γ*,ϱ,ϱ*,ω,η,η* taken from K such that both [Formula: see text] The sequence is uniquely determined by the Leonard pair provided the dimension of V is at least 4. The equations above are called the Askey–Wilson relations.

scholarly journals On Topological Semigroups of Matrix Units

2005 ◽  
Vol 71 (3) ◽  
pp. 389-400 ◽  
Author(s):  
Oleg V. Gutik ◽  
Kateryna P. Pavlyk
Keyword(s):  
Topological Semigroups ◽  
Matrix Units

scholarly journals On the generators of elementary subgroups of general linear groups

1996 ◽  
Vol 38 (1) ◽  
pp. 1-10 ◽  
Author(s):  
A. W. Mason
Keyword(s):  
Commutative Rings ◽  
General Linear ◽  
Linear Groups ◽  
General Linear Groups ◽  
Simple Set ◽  
Matrix Units

Let R be a ring with identity and let Eij ∈ Mn(R) be the usual n X n matrix units, where n ≥ 2 and 1≤i, j≤N. Let En(R) be the subgroup of GLn(R) generated by all Tij(q where r ∈ R and i ≄ j. For each (two-sided) R-ideal q let En(R, q) be the normal subgroup of En(R) generated by Tij(q), where q ∈ q. The subgroup En(R, q) plays an important role in the theory of GLn(R). For example, Vaserˇstein has proved that, for a larger class of rings (which includes all commutative rings), every subgroup S of GLn(R), when R ∈ and n≥3, contains the subgroup En(R, q0), where q0 is the R-ideal generated by αij, rαij-αjjr (i ≄ j, r ∈ R), for all (αij) ∈ S. (See [13, Theorem 1].) In addition Vaseršstein has shown that, for the same class of rings, En(R, q) has a simple set of generators when n ≥ 3. Let Ên(R, q) be the subgroup of En(R, q) generated by Tij(r)Tij(q)Tij(−r), where r ∈ R, q ∈ q. Then Ên(R, q) = En(R, q), for all q, when R ∈ and n ≥ 3.(See [13, Lemma 8].)

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