Characteristic Relations for Quantum Matrices

General algebraic properties of the algebras of vector fields over quantum linear groups $GL_q(N)$ and $SL_q(N)$ are studied. These quantum algebras appears to be quite similar to the classical matrix algebra. In particular, quantum analogues of the characteristic polynomial and characteristic identity are obtained for them. The $q$-analogues of the Newton relations connecting two different generating sets of central elements of these algebras (the determinant-like and the trace-like ones) are derived. This allows one to express the $q$-determinant of quantized vector fields in terms of their $q$-traces.Comment: 11 pages, latex, an important reference [16] added

Development of a genetic linkage map for Sharon goatgrass (Aegilops sharonensis) and mapping of a leaf rust resistance gene

Aegilops sharonensis (Sharon goatgrass), a diploid wheat relative, is known to be a rich source of disease resistance genes for wheat improvement. To facilitate the transfer of these genes into wheat, information on their chromosomal location is important. A genetic linkage map of Ae. sharonensis was constructed based on 179 F2 plants derived from a cross between accessions resistant (1644) and susceptible (1193) to wheat leaf rust. The linkage map was based on 389 markers (377 Diversity Arrays Technology (DArT) and 12 simple sequence repeat (SSR) loci) and was comprised of 10 linkage groups, ranging from 2.3 to 124.6 cM. The total genetic length of the map was 818.0 cM, with an average interval distance between markers of 3.63 cM. Based on the chromosomal location of 115 markers previously mapped in wheat, the four linkage groups of A, B, C, and E were assigned to Ae. sharonensis (Ssh) and homoeologous wheat chromosomes 6, 1, 3, and 2. The single dominant gene (designated LrAeSh1644) conferring resistance to leaf rust race THBJ in accession 1644 was positioned on linkage group A (chromosome 6Ssh) and was flanked by DArT markers wpt-9881 (at 1.9 cM distal from the gene) and wpt-6925 (4.5 cM proximal). This study clearly demonstrates the utility of DArT for genotyping uncharacterized species and tagging resistance genes where pertinent genomic information is lacking

Mapping spot blotch resistance genes in four barley populations

Bipolaris sorokiniana (teleomorph: Cochliobolus sativus) is the fungal pathogen responsible for spot blotch in barley (Hordeum vulgare L.) and occurs worldwide in warmer, humid growing conditions. Current Australian barley varieties are largely susceptible to this disease and attempts are being made to introduce sources of resistance from North America. In this study we have compared chromosomal locations of spot blotch resistance reactions in four North American two-rowed barley lines; the North Dakota lines ND11231-12 and ND11231-11 and the Canadian lines TR251 and WPG8412-9-2-1. Diversity Arrays Technology (DArT)-based PCR, expressed sequence tag (EST) and SSR markers have been mapped across four populations derived from crosses between susceptible parental lines and these four resistant parents to determine the location of resistance loci. Quantitative trait loci (QTL) conferring resistance to spot blotch in adult plants (APR) were detected on chromosomes 3HS and 7HS. In contrast, seedling resistance (SLR) was controlled solely by a locus on chromosome 7HS. The phenotypic variance explained by the APR QTL on 3HS was between 16 and 25% and the phenotypic variance explained by the 7HS APR QTL was between 8 and 42% across the four populations. The SLR QTL on 7HS explained between 52 to 64% of the phenotypic variance. An examination of the pedigrees of these resistance sources supports the common identity of resistance in these lines and indicates that only a limited number of major resistance loci are available in current two-rowed germplasm

Quantum Gravity and the Algebra of Tangles

In Rovelli and Smolin's loop representation of nonperturbative quantum gravity in 4 dimensions, there is a space of solutions to the Hamiltonian constraint having as a basis isotopy classes of links in R^3. The physically correct inner product on this space of states is not yet known, or in other words, the *-algebra structure of the algebra of observables has not been determined. In order to approach this problem, we consider a larger space H of solutions of the Hamiltonian constraint, which has as a basis isotopy classes of tangles. A certain algebra T, the ``tangle algebra,'' acts as operators on H. The ``empty state'', corresponding to the class of the empty tangle, is conjectured to be a cyclic vector for T. We construct simpler representations of T as quotients of H by the skein relations for the HOMFLY polynomial, and calculate a *-algebra structure for T using these representations. We use this to determine the inner product of certain states of quantum gravity associated to the Jones polynomial (or more precisely, Kauffman bracket).Comment: 16 pages (with major corrections

Representation-theoretic derivation of the Temperley-Lieb-Martin algebras

Explicit expressions for the Temperley-Lieb-Martin algebras, i.e., the quotients of the Hecke algebra that admit only representations corresponding to Young diagrams with a given maximum number of columns (or rows), are obtained, making explicit use of the Hecke algebra representation theory. Similar techniques are used to construct the algebras whose representations do not contain rectangular subdiagrams of a given size.Comment: 12 pages, LaTeX, to appear in J. Phys.

The Nakayama automorphism of the almost Calabi-Yau algebras associated to SU(3) modular invariants

We determine the Nakayama automorphism of the almost Calabi-Yau algebra A associated to the braided subfactors or nimrep graphs associated to each SU(3) modular invariant. We use this to determine a resolution of A as an A-A bimodule, which will yield a projective resolution of A.Comment: 46 pages which constitutes the published version, plus an Appendix detailing some long calculations. arXiv admin note: text overlap with arXiv:1110.454

A New Young Diagrammatic Method For Kronecker Products of O(n) and Sp(2m)

A new simple Young diagrammatic method for Kronecker products of O(n) and Sp(2m) is proposed based on representation theory of Brauer algebras. A general procedure for the decomposition of tensor products of representations for O(n) and Sp(2m) is outlined, which is similar to that for U(n) known as the Littlewood rules together with trace contractions from a Brauer algebra and some modification rules given by King.Comment: Latex, 11 pages, no figure

The induced representations of Brauer algebra and the Clebsch-Gordan coefficients of SO(n)

Induced representations of Brauer algebra $D_{f}(n)$ from $S_{f_{1}}\times S_{f_{2}}$ with $f_{1}+f_{2}=f$ are discussed. The induction coefficients (IDCs) or the outer-product reduction coefficients (ORCs) of $S_{f_{1}}\times S_{f_{2}}\uparrow D_{f}(n)$ with $f\leq 4$ up to a normalization factor are derived by using the linear equation method. Weyl tableaus for the corresponding Gel'fand basis of SO(n) are defined. The assimilation method for obtaining CG coefficients of SO(n) in the Gel'fand basis for no modification rule involved couplings from IDCs of Brauer algebra are proposed. Some isoscalar factors of $SO(n)\supset SO(n-1)$ for the resulting irrep $[\lambda_{1},~\lambda_{2},~ \lambda_{3},~\lambda_{4},\dot{0}]$ with $\sum\limits_{i=1}^{4}\lambda_{i}\leq .Comment: 48 pages latex, submitted to Journal of Phys.

On fusion algebra of chiral SU(N)kSU(N)_{k} models

We discuss some algebraic setting of chiral $SU(N)_{k}$ models in terms of the statistical dimensions of their fields. In particular, the conformal dimensions and the central charge of the chiral $SU(N)_{k}$ models are calculated from their braid matrices. Futhermore, at level K=2, we present the characteristic polynomials of their fusion matrices in a factored form.Comment: 11 pages, ioplpp