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.

Groups with Representations of Bounded Degree

1964 ◽  
Vol 16 ◽  
pp. 299-309 ◽  
Author(s):  
I. M. Isaacs ◽  
D. S. Passman
Keyword(s):  
Group Algebra ◽  
Finite Index ◽  
Abelian Subgroup ◽  
Discrete Group ◽  
Irreducible Representations ◽  
Complex Numbers ◽  
Bounded Degree ◽  
Normal Abelian Subgroup

Let G be a discrete group with group algebra C[G] over the complex numbers C. In (5) Kaplansky essentially proves that if G has a normal abelian subgroup of finite index n, then all irreducible representations of C[G] have degree ≤n. Our main theorem is a converse of Kaplansky's result. In fact we show that if all irreducible representations of C[G] have degree ≤n, then G has an abelian subgroup of index not greater than some function of n. (The degree of a representation of C[G] for arbitrary G is defined precisely in § 3.)

Brauer Characters and Grothendieck Rings

1975 ◽  
Vol 27 (5) ◽  
pp. 1025-1028 ◽  
Author(s):  
B. M. Puttaswamaiah
Keyword(s):  
Finite Order ◽  
Group Algebra ◽  
Irreducible Representations ◽  
Splitting Field ◽  
Finitely Generated ◽  
Grothendieck Ring ◽  
Primitive Idempotents ◽  
Brauer Characters ◽  
Grothendieck Rings ◽  
Special Case

Let G be a group of finite order g, A a splitting field of G of characteristic p (which may be 0) and R = AG the group algebra of G over A. In [2], the author studied some of the properties of the Grothendieck ring K(R) of the category of all finitely generated R-modules, and derived a number of consequences. This paper continues the study carried out in [2]. The study is concerned with the structure and irreducible representations of K(R). The ring K(R) is proved to be semisimple and the primitive idempotents of K(R) are explicitly constructed. If the ring K(R) is identified with the ‘algebra of representations', then Robinson's idempotent [3; 4; 5] follow from our description as a special case.

Theoretical studies in nuclear structure IV. Wave functions for the nuclear p -shell Part A. ‹ p n │ p n-1 p › fractional parentage coefficients

1951 ◽  
Vol 209 (1099) ◽  
pp. 502-524 ◽  
Keyword(s):  
Direct Method ◽  
Wave Functions ◽  
Irreducible Representations ◽  
Reciprocal Relation ◽  
Spin Coefficient ◽  
Linear Combinations ◽  
Orthogonal Representation ◽  
Fractional Parentage ◽  
Complete Set ◽  
A Charge

A complete set of wave functions is constructed for the whole of the nuclear p -shell (from p 3 to p 12 ). Following Racah, the wave functions for p n are expressed as linear combinations of totally antisymmetric wave functions for p n-1 , vector-coupled to the wave functions of the remaining particle. The coefficients in the linear combination are expressed as the product of an orbital coefficient, a charge-spin- coefficient and a weight factor equal to the square root of the ratio of the dimensions of two irreducible representations of permutation groups. Using the Young-Yamanouchi orthogonal representation of the permutation group, the orbital and charge-spin coefficients may be calculated independently. Specialization of the new method to the atomic p -shell and an alternative direct method of calculating the total parentage coefficients are described in the appendices. A reciprocal relation for the special unitary group, simplifying the calculation of both the orbital and the charge-spin coefficients, is described in an Addendum.

An equation of quantum mechanics

1947 ◽  
Vol 43 (3) ◽  
pp. 406-413 ◽  
Author(s):  
D. E. Littlewood
Keyword(s):  
Quantum Mechanics ◽  
Binomial Coefficient ◽  
Irreducible Representations ◽  
Matrix Representations ◽  
Complete Set ◽  
Explicit Representations ◽  
Image Position

The complete set of matrix representations of a set of n quantities β1, β2,…,βn satisfyingis obtained for all values of n. It is found that if n = 2v or n = 2v + 1, there are v + 1 irreducible representations with the respective degrees 1, but if n = 2v + 1 there are in addition two conjugate representations of order , the symbol denoting the binomial coefficient. The explicit representations are given for n = 2, 3, 4, 5.

THE RATIONAL GROUP ALGEBRA OF A FINITE GROUP

2012 ◽  
Vol 12 (03) ◽  
pp. 1250168 ◽  
Author(s):  
GURMEET K. BAKSHI ◽  
RAVINDRA S. KULKARNI ◽  
INDER BIR S. PASSI
Keyword(s):  
Finite Group ◽  
Explicit Expression ◽  
Group Algebra ◽  
Irreducible Character ◽  
Metabelian Group ◽  
Central Idempotent ◽  
Rational Group ◽  
Wedderburn Decomposition ◽  
Complete Set ◽  
A Finite Group

An explicit expression for the primitive central idempotent of the rational group algebra ℚ[G] of a finite group G associated with any complex irreducible character of G is obtained. A complete set of primitive central idempotents and the Wedderburn decomposition of the rational group algebra of a finite metabelian group is also computed.

ADE SUBALGEBRAS OF THE TRIPLET VERTEX ALGEBRA $\mathcal{W}(p)$: A-SERIES

2013 ◽  
Vol 15 (06) ◽  
pp. 1350028 ◽  
Author(s):  
DRAŽEN ADAMOVIĆ ◽  
XIANZU LIN ◽  
ANTUN MILAS
Keyword(s):  
Central Charge ◽  
Strong Support ◽  
Vertex Algebra ◽  
Finite Subgroup ◽  
Irreducible Representations ◽  
Finite Subgroups ◽  
Complete Set ◽  
Ade Classification ◽  
Charge Formula

Motivated by [On the triplet vertex algebra [Formula: see text], Adv. Math.217 (2008) 2664–2699], for every finite subgroup Γ ⊂ PSL(2, ℂ) we investigate the fixed point subalgebra [Formula: see text] of the triplet vertex [Formula: see text], of central charge [Formula: see text], p ≥ 2. This part deals with the A-series in the ADE classification of finite subgroups of PSL(2, ℂ). First, we prove the C2-cofiniteness of the Am-fixed subalgebra [Formula: see text]. Then we construct a family of [Formula: see text]-modules, which are expected to form a complete set of irreducible representations. As a strong support to our conjecture, we prove modular invariance of (generalized) characters of the relevant (logarithmic) modules. Further evidence is provided by calculations in Zhu's algebra for m = 2. We also present a rigorous proof of the fact that the full automorphism group of [Formula: see text] is PSL(2, ℂ).

scholarly journals Random Cayley Graphs are Expanders: a Simple Proof of the Alon–Roichman Theorem

10.37236/1815 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Zeph Landau ◽  
Alexander Russell
Keyword(s):  
Finite Group ◽  
Cayley Graph ◽  
Group Algebra ◽  
Simple Proof ◽  
Cayley Graphs ◽  
Random Variables ◽  
Irreducible Representations ◽  
Original Proof ◽  
A Finite Group ◽  
Second Eigenvalue

We give a simple proof of the Alon–Roichman theorem, which asserts that the Cayley graph obtained by selecting $c_\varepsilon \log |G|$ elements, independently and uniformly at random, from a finite group $G$ has expected second eigenvalue no more than $\varepsilon$; here $c_\varepsilon$ is a constant that depends only on $\varepsilon$. In particular, such a graph is an expander with constant probability. Our new proof has three advantages over the original proof: (i.) it is extremely simple, relying only on the decomposition of the group algebra and tail bounds for operator-valued random variables, (ii.) it shows that the $\log |G|$ term may be replaced with $\log D$, where $D \leq |G|$ is the sum of the dimensions of the irreducible representations of $G$, and (iii.) it establishes the result above with a smaller constant $c_\varepsilon$.

scholarly journals Construction of Codes from Symmetric Groups

2019 ◽  
Vol 8 (4) ◽  
pp. 8658-8665
Keyword(s):  
Symmetric Group ◽  
Group Algebra ◽  
Minimum Distance ◽  
Symmetric Groups ◽  
Semisimple Group ◽  
Group Codes ◽  
Complete Set

Let FSn be semisimple group algebra where Sn denotes the Symmetric group of degree n. We obtain the complete set of irreducible linear idempotents of the group algebra FSn. We also find the dimension and minimum distance of the group codes over the group S

scholarly journals REPRESENTATIONS OF THE RENNER MONOID

2009 ◽  
Vol 19 (04) ◽  
pp. 511-525 ◽  
Author(s):  
ZHUO LI ◽  
ZHENHENG LI ◽  
YOU'AN CAO
Keyword(s):  
Representation Theory ◽  
Crucial Role ◽  
Irreducible Representations ◽  
Weyl Groups ◽  
Parabolic Subgroups ◽  
Reductive Monoids ◽  
Complete Set ◽  
Character Formulas ◽  
Renner Monoid

We describe irreducible representations and character formulas of the Renner monoids for reductive monoids, which generalize the Munn–Solomon representation theory of rook monoids to any Renner monoids. The type map and polytope associated with reductive monoids play a crucial role in our work. It turns out that the irreducible representations of certain parabolic subgroups of the Weyl groups determine the complete set of irreducible representations of the Renner monoids. An analogue of the Munn–Solomon formula for calculating the character of the Renner monoids, in terms of the characters of the parabolic subgroups, is shown.

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