Character Tables Of Groups Research Articles

Row–column duality and combinatorial topological strings

Integrality properties of partial sums over irreducible representations, along columns of character tables of finite groups, were recently derived using combinatorial topological string theories (CTST). These CTST were based on Dijkgraaf-Witten theories of flat G-bundles for finite groups G in two dimensions, denoted G-TQFTs. We define analogous combinatorial topological strings related to two dimensional topological field theories (TQFTs) based on fusion coefficients of finite groups. These TQFTs are denoted as R(G)-TQFTs and allow analogous integrality results to be derived for partial row sums of characters over conjugacy classes along fixed rows. This relation between the G-TQFTs and R(G)-TQFTs defines a row-column duality for character tables, which provides a physical framework for exploring the mathematical analogies between rows and columns of character tables. These constructive proofs of integrality are complemented with the proof of similar and complementary results using the more traditional Galois theoretic framework for integrality properties of character tables. The partial row and column sums are used to define generalised partitions of the integer row and column sums, which are of interest in combinatorial representation theory.

Computing character tables of symmetric groups using higher order permutation representations

In this paper, we will introduce the topic of representation theory by focusing on the techniques of computing character tables of finite groups, in particular, the symmetric group Sn . First, we will introduce some basic concepts of representation theory of finite groups over ℂ. Then we will compute the character tables for S 3 and S 4 using the classical methods introduced in Section 2. Also, we will construct a series of representations for Sn and compute their characters. Finally, we will compute the character tables for S 5 and S 6 using our method introduced in Section 3.

Zeros in the character tables of symmetric groups with an -core index

AbstractLet $\mathcal {C}_n =\left [\chi _{\lambda }(\mu )\right ]_{\lambda , \mu }$ be the character table for $S_n,$ where the indices $\lambda $ and $\mu $ run over the $p(n)$ many integer partitions of $n.$ In this note, we study $Z_{\ell }(n),$ the number of zero entries $\chi _{\lambda }(\mu )$ in $\mathcal {C}_n,$ where $\lambda $ is an $\ell $ -core partition of $n.$ For every prime $\ell \geq 5,$ we prove an asymptotic formula of the form $$ \begin{align*}Z_{\ell}(n)\sim \alpha_{\ell}\cdot \sigma_{\ell}(n+\delta_{\ell})p(n)\gg_{\ell} n^{\frac{\ell-5}{2}}e^{\pi\sqrt{2n/3}}, \end{align*} $$ where $\sigma _{\ell }(n)$ is a twisted Legendre symbol divisor function, $\delta _{\ell }:=(\ell ^2-1)/24,$ and $1/\alpha _{\ell }>0$ is a normalization of the Dirichlet L-value $L\left (\left ( \frac {\cdot }{\ell } \right ),\frac {\ell -1}{2}\right ).$ For primes $\ell $ and $n>\ell ^6/24,$ we show that $\chi _{\lambda }(\mu )=0$ whenever $\lambda $ and $\mu $ are both $\ell $ -cores. Furthermore, if $Z^*_{\ell }(n)$ is the number of zero entries indexed by two $\ell $ -cores, then, for $\ell \geq 5$ , we obtain the asymptotic $$ \begin{align*}Z^*_{\ell}(n)\sim \alpha_{\ell}^2 \cdot \sigma_{\ell}( n+\delta_{\ell})^2 \gg_{\ell} n^{\ell-3}. \end{align*} $$

Characters of twisted fractional linear groups

We determine character tables for twisted fractional linear groups that form the “other” family in Zassenhaus’ classification of finite sharply 3-transitive groups.

On a maximal subgroup of the orthogonal group O⁺₈(3)

The orthogonal simple group 0 (3) has three conjugacy classes of maximal subgroups of the form 36:L4(3). These groups are all isomorphic to each other and each group has order 4421589120 with index 1120 in 0 (3). In this paper, we will compute the ordinary carácter table of one of these classes of maximal subgroups using the technique of Fischer-Clifford matrices. This technique is very efficient to compute the ordinary character table of an extension group Ḡ = N.G and especially where the normal subgroup N of Ḡ is an elementary abelian p-group. The said technique reduces the computation of the ordinary character table of Ḡ to find a handful of so-called Fischer-Clifford matrices of Ḡ and the ordinary or projective character tables of the inertia factor groups of the action of Ḡ on N.

Large absolute values of cyclotomic polynomials at roots of unity

The nth cyclotomic polynomial Φn(X) is the minimal polynomial of ζn:=e2πi/n. Given an integer m≥1 and a prescribed set S of arithmetic progressions modulo m, we define nx as the product of the primes p≤x lying in those progressions. Let d(n) denote the number of divisors of n. It turns out that under certain conditions on S and m there exists jx such that log⁡|Φnx(ζmjx)|/d(nx) tends to a positive limit. Our aim is to determine those conditions. We use the arithmetic of cyclotomic number fields, non-standard properties of character tables of finite abelian groups and a recent theorem of Bzdȩga, Herrera-Poyatos and Moree. After developing some generalities, we restrict to the case where m is a prime.Our motivation comes from a paper of Vaughan (1975). He studied the case where S={±2(mod5)} and used it to show that the maximum coefficient in absolute value of Φn can be very large.

The sparsity of character tables of high rank groups of Lie type

In the high rank limit, the fraction of non-zero character table entries of finite simple groups of Lie type goes to zero.

On the character tables of symmetric groups

In this paper, some zeros and non-zeros in the character tables of symmetric groups are displayed in the partition forms. In particular, more zeros of self conjugate partitions beside odd permutations are heavily investigated.

Group Geometric Algebra and the Standard Model

We show how to generalize the Weyl equation to include the Standard Model fermions and a dark matter fermion. The 2 × 2 complex matrices are a matrix ring R. A finite group G can be used to define a group algebra G[R] which is a generalization of the ring. For a group of size N, this defines N Weyl equations coupled by the group operation. We use the group character table to uncouple the equations by diagonalizing the group algebra. Using the full octahedral point symmetry group for G, our uncoupled Weyl equations have the symmetry of the Standard Model fermions plus a dark matter particle. We describe the symmetry properties of dark matter.

On divisibility by primes in columns of character tables of symmetric groups

For an arbitrary prime p, we prove that the proportion of entries divisible by p in certain columns of the character table of the symmetric group $$S_n$$ tends to 1 as $$n\rightarrow \infty $$. This is done by finding lower bounds on the number of k-cores for k large enough with respect to n.

Dense proportions of zeros in character values

Proportions of zeros in character tables of finite groups are dense in [0,1].

Turning weight multiplicities into Brauer characters

We describe two methods for computing p-modular Brauer character tables for groups of Lie type G(pf) in defining characteristic p, assuming that the ordinary character table of G(pf) is known, and the weight multiplicities of the corresponding algebraic group G are known for p-restricted highest weights.

Primitive idempotent tables of cyclic and constacyclic codes

For any lambda in GF(q)^{ast} a lambda -constacyclic code C^{n,q,lambda } : = ,langle g(x) rangle , of length n is a set of polynomials in the ring GF(q)[x]/x^{n} - lambda , which is generated by some polynomial divisor g(x) of x^{n} - lambda . In this paper a general expression is presented for the uniquely determined idempotent generator of such a code. In particular, if g(x): = (x^{n} - lambda) / P_{t}^{n,q,lambda } (x) , where P_{t}^{n,q,lambda } (x) is an irreducible factor polynomial of x^{n} - lambda , one obtains a so-called minimal or irreducible constacyclic code. The idempotent generator of a minimal code is called a primitive idempotent generating polynomial or, shortly, a primitive idempotent. It is proven that for any triple (n,q,lambda ) with (n,q) = 1 the set of primitive idempotents gives rise to an orthogonal matrix. This matrix is closely related to a table which shows some resemblance with irreducible character tables of finite groups. The cases lambda = 1 (cyclic codes) and lambda = - 1 (negacyclic codes), which show this resemblance most clearly, are studied in more detail. All results in this paper are extensions and generalizations of those in van Zanten (Des Codes Cryptogr 75:315–334, 2015).

On a Maximal Parabolic Subgroup of $$O^{+}_{8}(2)$$ O 8 + ( 2 )

The simple orthogonal group $$O^{+}_{8}(2)$$ has a maximal parabolic subgroup of the form $$\bar{G}=N{:}G$$ , where $$N=2^{6}$$ and $$G\cong A_{8}$$ . Using Atlas, we can see that $$O^{+}_{8}(2)$$ has three maximal subgroups of type $$2^6{:}A_{8}$$ . These three maximal subgroups are conjugate in the full automorphism group of $$O^{+}_{8}(2)$$ , namely $$O^{+}_{8}(2){:}S_{3}$$ . Thus, we have the modules $$M_{1}=2^{6}$$ , $$M_{2}=2^{6}$$ , and $$M_{3}=2^{6}$$ on which $$A_{8}$$ acts irreducibly (therefore, these three modules are isomorphic as $$O^{+}_{8}(2)$$ -module). Without loss of generality, we can consider our maximal parabolic subgroup to be $$\bar{G}=M_{1}{:}A_{8}$$ . The Fischer matrices for each class representative of G are computed which together with character tables of inertia factor groups of G lead to the full character table of $$\bar{G}$$ . The complete fusion of $$\bar{G}$$ into $$O^{+}_{8}(2)$$ has been determined using the technique of set intersections of characters.

Character tables of n-dimensional hyperoctahedral groups and their applications

ABSTRACTWe have shown that powerful matrix-type-based multinomial generators together with combinatorial techniques can be applied to derive the conjugacy classes, irreducible representations and the character tables of n-dimensional hyperoctahedral or hypercube groups of order n! × 2n, which are generalisations of non-rigid molecular theory formulated by Longuet-Higgins. The hyperoctahedral groups appear in the symmetry groups of non-rigid molecules, non-rigid water clusters, and other weakly bound van der Waals complexes that exhibit rapid tunnelling motion among various low-lying surmountable potential energy minima. These groups also have applications in quantum chromodynamics in the classification of states of quarks. The associated combinatorics has applications in enumerations, nuclear spin statistics and NMR spectroscopy. Finally, the groups enhance understanding of chirality in biosystems, that is, the functionality of intrinsically disordered proteins. We have derived the character tables of hyperoctahedral groups of sixth dimension, and techniques have been outlined to generalise for higher dimensions in the form of matrix generators.

On 2 nonsplit extension groups associated with HS and HS:2

The group HS:2 is the full automorphism group of the Higman--Sims group HS. The groups 2^{4.}S_6 and 2^{5.}S_6 are maximal subgroups of HS and HS:2, respectively. The group 2^{4.}S_6 is of order 11520 and 2^{5.}S_6 is of order 23040 and each of them is of index 3 850 in HS and HS:2, respectively. The aim of this paper is to first construct \overline{G} = 2^{5.}S_6 as a group of the form 2^{4.}S_6.2 (that is, \overline{G} = G_1.2) and then compute the character tables of these 2 nonsplit extension groups by using the method of Fischer--Clifford theory. We will show that the projective character tables of the inertia factor groups are not required. The Fischer--Clifford matrices of \overline{G}_1 and \overline{G} are computed. These matrices together with the partial character tables of the inertia factors are used to compute the full character tables of these 2 groups. The fusion of \overline{G}_1 into \overline{G} is also given.

The T-matrix code Tsym for homogeneous dielectric particles with finite symmetries

A T-matrix code tailored to non-axisymmetric particles with finite symmetries is described. The code exploits geometric symmetries of particles by use of group theoretical methods. Commutation relations of the T-matrix are implemented for reducing CPU-time requirements. Irreducible representations of finite groups are employed for alleviating ill-conditioning problems in numerical computations. Further, an iterative T-matrix method for particles with small-scale surface perturbations is implemented. The code can compute both differential and integrated optical properties of particles in either fixed or random orientation. Methods for testing the convergence and correctness of the computational results are discussed. The package also includes a database of pre-computed group-character tables, as well as an interface to the GAP programming language for computational group theory. The code can be downloaded at http://www.rss.chalmers.se/∼kahnert/Tsym.html.

Submatrices of character tables and basic sets

In this investigation of character tables of finite groups we study basic sets and associated representation theoretic data for complementary sets of conjugacy classes. For the symmetric groups we find unexpected properties of characters on restricted sets of conjugacy classes, like beautiful combinatorial determinant formulae for submatrices of the character table and Cartan matrices with respect to basic sets; we observe that similar phenomena occur for the transition matrices between power sum symmetric functions to bounded partitions and the k-Schur functions defined by Lapointe and Morse. Arithmetic properties of the numbers occurring in this context are studied via generating functions.

Symmetry effects on the optical coupling between plasmonic nanoparticles with applications to hierarchical structures

We present theoretical studies on the collective optical response of symmetric configurations of metallic nanoparticles. We show that within the electrostatic approximation, the surface plasmon resonance of these symmetric multiparticle systems can be expressed as symmetry-adapted linear combinations of the plasmon modes of each particle of the ensemble, closely resembling the situation encountered in molecular systems. By making use of group theoretical arguments, we show that such linear combinations can be written down by simple geometrical considerations through the use of point group character tables, without using extensive numerical computations. Furthermore, we apply this formalism to study the coupling of hierarchical arrays containing a large number of nanoparticles. This theory thus provides an intuitive and formal approach for the rational design of plasmonic nanostructures.

Wyckoff positions and the expression of polarization singularities in photonic crystals

We reveal the fundamental relation between linear photonic crystal symmetries and the local polarization states of its Bloch modes, in particular the location and nature of polarization singularities as established by rigorous group theoretic analysis, encompassing the full system symmetry. This is illustrated with the fundamental transverse electric mode of a two-dimensional hexagonal photonic crystal, in the vanishing contrast limit and at the K point. For general Wyckoff positions within the fundamental domain, the transformation of a local polarization state is determined by the nature of the symmetry operations that map to members of its crystallographic orbit. In particular, the site symmetries that correspond to specific Wyckoff positions constrain the local polarization state to singular character--circular, linear or disclination. Moreover, through the application of a local symmetry transformation relation, and the group's character table, the precise natures of the singularities may be determined from self-consistency arguments.