Character Tables Research Articles

Efficient molecular encoding in multifunctional self-immolative urethanes

SUMMARYMolecular encoding in sequence-defined polymers shows promise as a new paradigm for data storage. Here, we report what is, to our knowledge, the first use of self-immolative oligourethanes for storing and reading encoded information. As a proof of principle, we describe how a text passage from Jane Austen’s Mansfield Park was encoded in sequence-defined oligourethanes and reconstructed via self-immolative sequencing. We develop Mol.E-coder, a software tool that uses a Huffman encoding scheme to convert the character table to hexadecimal. The oligourethanes are then generated by a high-throughput parallel synthesis. Sequencing of the oligourethanes by self-immolation is done concurrently in a parallel fashion, and the liquid chromatography-mass spectrometry (LC-MS) information decoded by our Mol.E-decoder software. The passage is capable of being reproduced wholly intact by a third-party, without any purifications or the use of tandem MS (MS/MS), despite multiple rounds of compression, encoding, and synthesis.

Average number of zeros of characters of finite groups

There has been some interest on how the average character degree affects the structure of a finite group. We define, and denote by anz(G), the average number of zeros of characters of a finite group G as the number of zeros in the character table of G divided by the number of irreducible characters of G. We show that if anz(G)<1, then the group G is solvable and also that if anz(G)<12, then G is supersolvable. We characterise abelian groups by showing that anz(G)<13 if and only if G is abelian.

Landau's theorem and the number of conjugacy classes of zeros of characters

Motivated by a 2004 conjecture by the author and J. Sangroniz, Y. Yang has recently proved that if G is solvable then the index in G of the 8th term of the ascending Fitting series is bounded in terms of the largest number of zeros in a row in the character table of G. In this note, we prove this result for arbitrary finite groups and propose a stronger form of the 2004 conjecture. We conclude the paper showing some possible ways to prove this strengthened conjecture.

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.

English

Let G be a finite group. If G has two rows which differ in only one entry in the character table, we call G an RD1-group. We investigate the character tables of RD1-groups and get some necessary and sufficient conditions about RD1-groups.

Integral Cayley graphs over semi-dihedral groups

Classifying integral graphs is a hard problem that initiated by Harary and Schwenk in 1974. In this paper, with the help of character table, we treat the corresponding problem for Cayley graphs over the semi-dihedral group SD8n = ?a,b | a4n = b2 = 1, bab = a2n-1?, n ? 2. We present several necessary and sufficient conditions for the integrality of Cayley graphs over SD8n, we also obtain some simple sufficient conditions for the integrality of Cayley graphs over SD8n in terms of the Boolean algebra of hai. In particular, we give the sufficient conditions for the integrality of Cayley graphs over semi-dihedral groups SD2n (n?4) and SD8p for a prime p, from which we determine several infinite classes of integral Cayley graphs over SD2n and SD8p.

Finite Groups for the Kummer Surface: The Genetic Code and a Quantum Gravity Analogy

The Kummer surface was constructed in 1864. It corresponds to the desingularization of the quotient of a 4-torus by 16 complex double points. Kummer surface is known to play a role in some models of quantum gravity. Following our recent model of the DNA genetic code based on the irreducible characters of the finite group G5:=(240,105)≅Z5⋊2O (with 2O the binary octahedral group), we now find that groups G6:=(288,69)≅Z6⋊2O and G7:=(336,118)≅Z7⋊2O can be used as models of the symmetries in hexamer and heptamer proteins playing a vital role for some biological functions. Groups G6 and G7 are found to involve the Kummer surface in the structure of their character table. An analogy between quantum gravity and DNA/RNA packings is suggested.

Finding supercharacter theories on character tables

We describe an easy way how to find supercharacter theories for a finite group, if its character table is known. Namely, we show how an arbitrary partition of the conjugacy classes or of the irreducible characters can be refined to the coarsest partition that belongs to a supercharacter theory. Our constructions emphasize the duality between superclasses and supercharacters. An algorithm is presented to find all supercharacter theories on a given character table. The algorithm is used to compute the number of supercharacter theories for some nonabelian simple groups with up to 26 conjugacy classes.

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.

On a two-fold cover 2.(2⁶˙G₂(2)) of a maximal subgroup of Rudvalis group Ru

The Schur multiplier M(Ḡ1) ≅4 of the maximal subgroup Ḡ1 = 2⁶˙G₂(2)of the Rudvalis sporadic simple group Ru is a cyclic group of order 4. Hence a full representative group R of the type R = 4.(2⁶˙G₂(2)) exists for Ḡ1. Furthermore, Ḡ1 will have four sets IrrProj(Ḡ1;αi) of irreducible projective characters, where the associated factor sets α1, α2, α3 and α4, have orders of 1, 2, 4 and 4, respectively. In this paper, we will deal with a 2-fold cover 2. Ḡ1 of Ḡ1 which can be treated as a non-split extension of the form Ḡ = 27˙G2(2). The ordinary character table of Ḡ will be computed using the technique of the so-called Fischer matrices. Routines written in the computer algebra system GAP will be presented to compute the conjugacy classes and Fischer matrices of Ḡ and as well as the sizes of the sets |IrrProj(Hi; αi)| associated with each inertia factor Hi. From the ordinary irreducible characters Irr(Ḡ) of Ḡ, the set IrrProj(Ḡ1; α2) of irreducible projective characters of Ḡ1 with factor set α2 such that α22= 1, can be obtained.

Real constituents of permutation characters

We prove a broad generalization of a theorem of W. Burnside about the existence of real characters of finite groups to permutation characters. If G is a finite group, under the necessary hypothesis of O2′(G)=G, we can also give some control on the parity of multiplicities of the constituents of permutation characters (a result that needs the Classification of Finite Simple Groups). Along the way, we give a new characterization of the 2-closed finite groups using odd-order real elements of the group. All this can be seen as a contribution to Brauer's Problem 11 which asks how much information about subgroups of a finite group can be determined by the character table.

The Fitting subgroup, p‐length, derived length and character table

AbstractFor a character χ of a finite group G, the number is called the codegree of χ. Let N be a normal subgroup of G and set Let p be a prime. In this paper, we first show that if for two distinct prime divisors p and q of , divides none of the codegrees of elements of , then and N is either p‐solvable or q‐solvable. Next, we classify the finite groups with exactly one irreducible character of the codegree divisible by p and, also finite groups whose codegrees of irreducible characters which are divisible by p are equal. Then, we prove that p‐length of a finite p‐solvable group is not greater than the number of the distinct codegrees of its irreducible characters which are divisible by p. Finally, we consider the case when the codegree of every element of is square‐free.

A Compression System for Unicode Files Using an Enhanced Lzw Method

Data compression plays a vital and pivotal role in the process of computing as it helps in space reduction occupied by a file as well as to reduce the time taken to access the file. This work relates to a method for compressing and decompressing a UTF-8 encoded stream of data pertaining to Lempel-Ziv-welch (LZW) method. It is worth to use an exclusive-purpose LZW compression scheme as many applications are utilizing Unicode text. The system of the present work comprises a compression module, configured to compress the Unicode data by creating the dictionary entries in Unicode format. This is accomplished with adaptive characteristic data compression tables built upon the data to be compressed reflecting the characteristics of the most recent input data. The decompression module is configured to decompress the compressed file with the help of unique Unicode character table obtained from the compression module and the encoded output. We can have remarkable gain in compression, wherein the knowledge that we gather from the source is used to explore the decompression process.

Group theory Character Table Enhancement by Introducing the Partial Molecular Symmetry Principle in the Molecular Spectroscopy

Group theory Character Table Enhancement by Introducing the Partial Molecular Symmetry Principle in the Molecular Spectroscopy

A taxonomic revision of the Epischoenus group of Schoenus (Cyperaceae, tribe Schoeneae)

A taxonomic revision of the Epischoenus group of the southern African Schoenus is presented. Based on data from a review of specimens from nine herbaria and recent field collections, we revise the delimitations and distributions of eight species (Schoenus adnatus, Schoenus complanatus, Schoenus dregeanus, Schoenus gracillimus, Schoenus lucidus, Schoenus neovillosus, Schoenus quadrangularis and Schoenus schonlandii). In addition, we describe three new species to science: Schoenus crinitus, Schoenus rigidus and Schoenus selinae—all endemic to the Western Cape Province and adjacent areas of the Eastern Cape Province of South Africa. For these 11 Schoenus species, we update nomenclature, species circumscriptions and distribution maps, as well as present character tables and provide a taxonomic key. Species in the Epischoenus and allies group are perennial, caespitose graminoids that generally lack leaves along the lower parts of their culms, and they all have a conspicuous prolongation of the internodes between the florets of each spikelet. Differences in culm and inflorescence shape, hairiness and nutlet form are key characters that can be used to differentiate the species within this group.

Characters of Renner monoids and their Hecke algebras

This paper gives a general algorithm for computing the character table of any Renner monoid Hecke algebra, by adapting and generalizing techniques of Solomon used to study the rook monoid. The character table of the Hecke algebra of the rook monoid (i.e. the Cartan type [Formula: see text] Renner monoid) was computed earlier by Dieng et al. [2], using different methods. Our approach uses analogues of so-called A- and B-matrices of Solomon. In addition to the algorithm, we give explicit combinatorial formulas for the A- and B-matrices in Cartan type [Formula: see text] and use them to obtain an explicit description of the character table for the type [Formula: see text] Renner monoid Hecke algebra.

A revision of the genus Psammogorgia Verrill, 1868 (Cnidaria, Anthozoa, Octocorallia) in the tropical eastern Pacific Ocean.

The species of the genus Psammogorgia Verrill, 1868 from the shallow waters of the tropical eastern Pacific were mainly described from 1846 to 1870. Very few contributions were published subsequently. Recently, the genus was revisited with the addition of two new species. However, a comprehensive generic study is still missing for the eastern Pacific. Psammogorgia is characterised by having axes cores without mineralisation, mainly coarse irregular spindles and thorny, leafy or tuberculate clubs coenenchymal sclerites and the anthocodial armature with distinct collaret and points arrangements. Herein a taxonomic revision of the genus is presented based on type material which was morphologically analysed and illustrated using optical and scanning electron microscopy. Comparative character tables are provided for comparison among species in the genus, along with a taxonomic key. Moreover, the taxonomic status of each species was analysed. The genus Psammogorgia comprises six valid species and two varieties, and three lectotypes and a new combination are proposed to establish the taxonomic status of these species.

Character tables and defect groups

Let B be a block of a finite group G with defect group D. We prove that the exponent of the center of D is determined by the character table of G. In particular, we show that D is cyclic if and only if B contains a “large” family of irreducible p-conjugate characters. More generally, for abelian D we obtain an explicit formula for the exponent of D in terms of character values. In small cases even the isomorphism type of D is determined in this situation. Moreover, it can read off from the character table whether |D/D′|=4 where D′ denotes the commutator subgroup of D. We also propose a new characterization of nilpotent blocks in terms of the character table.

Informationally Complete Characters for Quark and Lepton Mixings

A popular account of the mixing patterns for the three generations of quarks and leptons is through the characters κ of a finite group G. Here, we introduce a d-dimensional Hilbert space with d = c c ( G ) , the number of conjugacy classes of G. Groups under consideration should follow two rules, (a) the character table contains both two- and three-dimensional representations with at least one of them faithful and (b) there are minimal informationally complete measurements under the action of a d-dimensional Pauli group over the characters of these representations. Groups with small d that satisfy these rules coincide in a large part with viable ones derived so far for reproducing simultaneously the CKM (quark) and PNMS (lepton) mixing matrices.

Classroom Friendly Method to Calculate Character Tables for Finite Groups

In this article, we show a very simple algorithm for the evaluation of the character tables applicable to all finite point groups. The method uses linear combinations of the symmetry operators associated with the point group’s classes as the basis set. Eigenvectors of the resulting matrix of such representations are related to the character table of the point group. The method is readily adaptable to both classroom demonstrations, exercises, and program implementation.