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We impose the periodicity conditions corresponding to the Matsubara formalism for Thermal Field Theory as constraints in the imaginary time path integral. These constraints are introduced by means of time-independent auxiliary fields which, by integration of the original variables, become dynamical fields in the resulting `dual' representation for the theory. This alternative representation has the appealing property of involving fields which live in one dimension less than the original ones, with a quantum partition function whose integration measure is identical to the one of its classical counterpart, albeit with a different (spatially nonlocal) action.

In this paper, we study the quantum $\mathfrak{sl}(n)$ representation category using the web space. Specially, we extend $\mathfrak{sl}(n)$ web space for $n\ge 4$ as generalized Temperley-Lieb algebras. As an application of our study, we find that the HOMFLY polynomial $P_n(q)$ specialized to a one variable polynomial can be computed by a linear expansion with respect to a presentation of the quantum representation category of $\mathfrak{sl}(n)$. Moreover, we correct the false conjecture \cite{PS:superiod} given by Chbili, which addresses the relation between some link polynomials of a periodic link and its factor link such as Alexander polynomial $(n = 0)$ and Jones polynomial $(n = 2)$ and prove the corrected conjecture not only for HOMFLY polynomial but also for the colored HOMFLY polynomial specialized to a one variable polynomial.

In this paper, we study the quantum $\mathfrak{sl}(n)$ representation category using the web space. Specially, we extend $\mathfrak{sl}(n)$ web space for $n\ge 4$ as generalized Temperley-Lieb algebras. As an application of our study, we find that the HOMFLY polynomial $P_n(q)$ specialized to a one variable polynomial can be computed by a linear expansion with respect to a presentation of the quantum representation category of $\mathfrak{sl}(n)$. Moreover, we correct the false conjecture \cite{PS:superiod} given by Chbili, which addresses the relation between some link polynomials of a periodic link and its factor link such as Alexander polynomial $(n = 0)$ and Jones polynomial $(n = 2)$ and prove the corrected conjecture not only for HOMFLY polynomial but also for the colored HOMFLY polynomial specialized to a one variable polynomial.

The paper is devoted to the study of some well-knonw combinatorial functions on the symmetric group $\sn$ --- the major index $\maj$, the descent number $\des$, and the inversion number $\inv$ --- from the representation-theoretic point of view. We show that each of these functions generates in the group algebra the same ideal, and the restriction of the left regular representation to this ideal is isomorphic to the representation of $\sn$ in the space of $n\times n$ skew-symmetric matrices. This allows us to obtain formulas for the functions $\maj,\des,\inv$ in terms of matrices of an exceptionally simple form. These formulas are applied to find the spectra of the elements under study in the regular representation, as well as to deduce a series of identities relating these functions to one another and to the number of fixed points $\fix$. \smallskip\noindent {\it Keywords:} major index, descent number, inversion number, representations of the symmetric group, skew-symmetric matrices, dual complexity.

We describe the generators and prove a number of relations for the construction of a planar algebra from the restricted quantum group $\bar{U}_{q}(\mathfrak{sl}_{2})$. This is a diagrammatic description of $End_{\bar{U}_{q}(\mathfrak{sl}_{2})}(X^{\otimes n})$, where $X:=\mathcal{X}^{+}_{2}$ is a two dimensional $\bar{U}_{q}(\mathfrak{sl}_{2})$ module.

In this paper, we present an exact procedure to deal with Dirac strings or worldsheets in gauge theories containing ensembles of monopoles interacting with charged fields. For SU(2) Yang-Mills theory, initially we construct the appropriate change of variables of the charged fields (including charged ghosts and auxiliary fields) so that the only change in the integrand of the partition function, in the Maximal Abelian gauge, is the addition of given closed Dirac worldsheets. Next, we derive our main result, namely, we show that it is always possible to choose them in such a manner that the total (open plus closed) Dirac worldsheets explicitly decouple from the charged sector, leaving only the effect of their associated gauge invariant borders (where the monopoles are placed), without missing any information about the center vortex sector. This procedure serves as a simplifying basis to deal with ensembles of monopoles and center vortices in the framework of the Cho-Faddeev-Niemi gauge field decomposition, by writing the partition function only in terms of the physical part of the defects to be integrated.

In this paper we prove theorems that describe how the representation theory of the affine Hecke algebra of type A and of related algebras such as the group algebra of the symmetric group are controlled by integrable highest weight representations of the characteristic zero affine Lie algebra \hat{sl}_l. In particular we parameterise the representations of these algebras by the nodes of the crystal graph, and give various Hecke theoretic descriptions of the edges. As a consequence we find for each prime p a basis of the integrable representations of \hat{sl}_l which shares many of the remarkable properties, such as positivity, of the global crystal basis/canonical basis of Lusztig and Kashiwara. This {\it $p$-canonical basis} is the usual one when p = 0, and the crystal of the p-canonical basis is always the usual one. The paper is self-contained, and our techniques are elementary (no perverse sheaves or algebraic geometry is invoked).

By exploiting the properties of q-deformed Coxeter elements, the scattering matrices of affine Toda field theories with real coupling constant related to any dual pair of simple Lie algebras may be expressed in a completely generic way. We discuss the governing equations for the existence of bound states, i.e. the fusing rules, in terms of q-deformed Coxeter elements, twisted q-deformed Coxeter elements and undeformed Coxeter elements. We establish the precise relation between these different formulations and study their solutions. The generalized S-matrix bootstrap equations are shown to be equivalent to the fusing rules. The relation between different versions of fusing rules and quantum conserved quantities, which result as nullvectors of a doubly q-deformed Cartan like matrix, is presented. The properties of this matrix together with the so-called combined bootstrap equations are utilised in order to derive generic integral representations for the scattering matrix in terms of quantities of either of the two dual algebras. We present extensive case-by-case data, in particular on the orbits generated by the various Coxeter elements.

The authors show that a wide class of Fredholm determinants arising in the representation theory of "big" groups such as the infinite-dimensional unitary group, solve Painleve equations. Their methods are based on the theory of integrable operators and the theory of Riemann-Hilbert problems.

We study the structure and representations of a family of vertex algebras obtained from affine superalgebras by quantum reduction. As an application, we obtain in a unified way free field realizations and determinant formulas for all superconformal algebras.

String and particle braiding statistics are examined in a class of topological orders described by discrete gauge theories with a gauge group $G$ and a 4-cocycle twist $\omega_4$ of $G$'s cohomology group $\mathcal{H}^4(G,\mathbb{R}/\mathbb{Z})$ in 3 dimensional space and 1 dimensional time (3+1D). We establish the topological spin and the spin-statistics relation for the closed strings, and their multi-string braiding statistics. The 3+1D twisted gauge theory can be characterized by a representation of a modular transformation group SL$(3,\mathbb{Z})$. We express the SL$(3,\mathbb{Z})$ generators $\mathsf{S}^{xyz}$ and $\mathsf{T}^{xy}$ in terms of the gauge group $G$ and the 4-cocycle $\omega_4$. As we compactify one of the spatial directions $z$ into a compact circle with a gauge flux $b$ inserted, we can use the generators $\mathsf{S}^{xy}$ and $\mathsf{T}^{xy}$ of an SL$(2,\mathbb{Z})$ subgroup to study the dimensional reduction of the 3D topological order $\mathcal{C}^{3\text{D}}$ to a direct sum of degenerate states of 2D topological orders $\mathcal{C}_b^{2\text{D}}$ in different flux $b$ sectors: $\mathcal{C}^{3\text{D}} = \oplus_b \mathcal{C}_b^{2\text{D}}$. The 2D topological orders $\mathcal{C}_b^{2\text{D}}$ are described by 2D gauge theories of the group $G$ twisted by the 3-cocycles $\omega_{3(b)}$, dimensionally reduced from the 4-cocycle $\omega_4$. We show that the SL$(2,\mathbb{Z})$ generators, $\mathsf{S}^{xy}$ and $\mathsf{T}^{xy}$, fully encode a particular type of three-string braiding statistics with a pattern that is the connected sum of two Hopf links. With certain 4-cocycle twists, we discover that, by threading a third string through two-string unlink into three-string Hopf-link configuration, Abelian two-string braiding statistics is promoted to non-Abelian three-string braiding statistics.

We apply some recent developments of Baldoni-Beck-Cochet-Vergne on vector partition function, to Kostant's and Steinberg's formulae, for classical Lie algebras $A\_r$, $B\_r$, $C\_r$, $D\_r$. We therefore get efficient {\tt Maple} programs that compute for these Lie algebras: the multiplicity of a weight in an irreducible finite-dimensional representation; the decomposition coefficients of the tensor product of two irreducible finite-dimensional representations. These programs can also calculate associated Ehrhart quasipolynomials.

A representation theory of the quantized Poincar\'e ($\kappa$-Poincar\'e) algebra (QPA) is developed. We show that the representations of this algebra are closely connected with the representations of the non-deformed Poincar\'e algebra. A theory of tensor operators for QPA is considered in detail. Necessary and sufficient conditions are found in order for scalars to be invariants. Covariant components of the four-momenta and the Pauli-Lubanski vector are explicitly constructed.These results are used for the construction of some q-relativistic equations. The Wigner-Eckart theorem for QPA is proven.

In his book "Cubic forms" Manin discovered that del Pezzo surfaces are related to root systems. To explain the many numerical coincidences Batyrev conjectured that a universal torsor on a del Pezzo surface can be embedded in a certain projective homogeneous space of the semisimple group with the same root system, equivariantly with respect to the maximal torus action. We prove this conjecture for del Pezzo surfaces of degrees greater than 1. Our proof uses an inductive process based on representations of Lie algebras corresponding to Hermitian symmetric pairs.

(Revised LaTex version).The structure of free field representation and some correlation functions of the SU(3) CFT are considered.

We first formulate a definition of tensor product for two modules for a vertex operator algebra in terms of a certain universal property and then we give a construction of tensor products. We prove the unital property of the adjoint module and the commutativity of tensor products, up to module isomorphism. We relate this tensor product construction with Frenkel and Zhu's $A(M)$-theory. We give a proof of a formula of Frenkel and Zhu for fusion rules. We also give the analogue of the ``Hom''-functor of classical Lie algebra theory for vertex operator algebra theory by introducing a notion of ``generalized intertwining operator.'' We prove that the space of generalized intertwining operators from one module to another for a vertex operator algebra is a generalized module. From this result we derive a general form of Tsuchiya and Kanie's ``nuclear democracy theorem'' for any rational vertex operator algebra. This proves that the fusion rules obtained from our construction of tensor products are the same as the fusion rules obtained by using Tsuchiya and Kanie's method, for both WZW models and minimal models. We prove that if $V$ satisfies certain ``finiteness'' and ``semisimplicity'' conditions, then there exists a unique maximal submodule inside the generalized module. Furthermore, we prove that this maximal submodule is isomorphic to the contragredient module of a certain tensor product module. This gives another construction of tensor product modules and this result turns out to be closely related to Huang and Lepowsky's construction.

In this section, we begin a systematic development of representation theory of �nite groups.

In this section, we begin a systematic development of representation theory of �nite groups.

-homogenous spaces with a discrete stability group; representations of the group of unimodular matrices of order 2 with elements from a locally compact topological field; representations of adele groups; appendixes-

-homogenous spaces with a discrete stability group; representations of the group of unimodular matrices of order 2 with elements from a locally compact topological field; representations of adele groups; appendixes-

We determine the decomposition numbers of the partition algebra when the characteristic of the ground field is zero or larger than the degree of the partition algebra. This will allow us to determine for which exact values of the parameter the partition algebras are semisimple over an arbitrary field. Furthermore, we show that the blocks of the partition algebra over an arbitrary field categorify weight spaces of an action of type A quantum groups on an analogue of the Fock space. In particular, we recover the block structure which was recently determined by Bowman et al. In order to do so, we use induction and restriction functors as well as analogues of Jucys-Murphy elements. The description of decomposition numbers will be in terms of combinatorics of partitions but can also be given a Lie theoretic interpretation in terms of a Weyl group of type A: A simple module $L(\mu)$ is a composition factor of a cell module $\Delta(\lambda)$ if and only if $\lambda$ and $\mu$ differ by the action of a transposition.

We review some recent advances in modular representation theory of symmetric groups and related Hecke algebras. We discuss connections with Khovanov-Lauda-Rouquier algebras and gradings on the blocks of the group algebras $F\Sigma_n$, which these connections reveal; graded categorification and connections with quantum groups and crystal bases; modular branching rules and the Mullineaux map; graded cellular structure and graded Specht modules; cuspidal systems for affine KLR algebras and imaginary Schur-Weyl duality, which connects representation theory of these algebras to the usual Schur algebras of smaller rank.

We give an alternate presentation of the cyclotomic rational Cherednik algebra, which has the useful feature of compatibility with the Opdam-Dunkl subalgebra. This presentation has a diagrammatic flavor, and it provides a simple explanation of several surprising facts about this algebra. It allows direct proof of the connection of category $\mathcal{O}$ to weighted KLR algebras, provides an algebraic construction of the KZ functor, and provides a better framework for connecting Cherednik algebras to Coulomb branches of 3-d gauge theories.

We study the automorphism group of the algebraic closure of a substructure A of a pseudo-finite field F, or more generally, of a bounded PAC field F. This paper answers some of the questions of [1], and in particular that any finite group which is geometrically represented in a pseudo-finite field must be abelian.

There has been substantial calculational progress in the last few years for gauge theory amplitudes which involve massless four dimensional particles. One of the central ingredients in this has been the ability to keep precise track of the Poincare algebra quantum numbers of the particles involved. Technically, this is most easily done using the well-known four dimensional spinor helicity method. In this article a natural generalization to all dimensions higher than four is obtained based on a covariant version of the representation theory of the Poincare algebra. Covariant expressions for all possible polarization states, both bosonic and fermionic, are constructed. For the fermionic states the analysis leads directly to pure spinors. The natural extension to the representation theory of the on-shell supersymmetry algebra results in an elementary derivation of the supersymmetry Ward identities for scattering amplitudes with massless or massive legs in any integer dimension from four onwards. As a proof-of-concept application a higher dimensional analog of the vanishing helicity-equal amplitudes in four dimensions is presented in (super) Yang-Mills theory, Einstein (super-)gravity and superstring theory in a flat background.

Looking for a quantum field theory model of Archimedean algebraic geometry a class of infinite-dimensional integral representations of classical special functions was introduced. Precisely the special functions such as Whittaker functions and Gamma-function were identified with correlation functions in topological field theories on a two-dimensional disk. Mirror symmetry of the underlying topological field theory leads to a dual finite-dimensional integral representations reproducing classical integral representations for the corresponding special functions. The mirror symmetry interchanging infinite- and finite-dimensional integral representations provides an incarnation of the local Archimedean Langlands duality on the level of classical special functions. In this note we provide some directions to higher-dimensional generalizations of our previous results. In the first part we consider topological field theory representations of multiple local L-factors introduced by Kurokawa and expressed through multiple Barnes's Gamma-functions. In the second part we are dealing with generalizations based on consideration of topological Yang-Mills theories on non-compact four-dimensional manifolds. Presumably, in analogy with the mirror duality in two-dimensions, S-dual description should be instrumental for deriving integral representations for a particular class of quantum field theory correlation functions and thus providing a new interesting class of special functions supplied with canonical integral representations.

In this paper, we study 2-representations of 2-quantum groups (in the sense of Rouquier and Khovanov-Lauda) categorifying tensor products of irreducible representations. Our aim is to construct knot homologies categorifying Reshetikhin-Turaev invariants of knots for arbitrary representations, which will be done in a follow-up paper. We consider an algebraic construction of these categories, via an explicit diagrammatic presentation, generalizing the cyclotomic quotient of the quiver Hecke algebra. One of our primary results is that these categories coincide when both are defined. We also investigate finer structure of these categories. Like many similar representation-theoretic categories, they are standardly stratified and satisfy a double centralizer property with respect to their self-dual modules. The standard modules of the stratification play an important role, as Vermas do in more classical representation theory, as test objects for functors. The existence of these representations has consequences for the structure of previously studied categorifications; it allows us to prove the non-degeneracy of Khovanov and Lauda's 2-category (that its Hom spaces have the expected dimension) in all symmetrizable types, and that the cyclotomic quiver Hecke algebras are symmetric Frobenius.

We perform a matching of the two loop-chiral perturbation theory representation of the scalar Kpi form factor to a dispersive one. Knowing the value of F_K/F_pi and f_+(0) in the Standard Model (SM) allows to determine two O(p^6) LECs, the slope of the scalar form factor and the deviation of the Callan-Treiman theorem. Going beyond the SM and assuming the knowledge of the slope of the scalar form factor from experiment, the matching allows us to determine the ratio of F_K/F_pi, f_+(0), a certain combination of non-standard couplings, the deviation of the Callan-Treiman theorem and the two O(p^6) LECs.

The following questions, concerning the application of the harmonic oscillator representation (HOR) in the theory of scattering and reactions, are discussed: the formulation of the scattering theory in HOR; exact solutions of the free motion Schroedinger equation in HOR; separable expansion of the short range potentials and the calculation of the phase shifts; 'isolated states' as generalization of the Wigner-von Neumann bound states embedded in continuum; a nuclear coupled channel problem in HOR; and the description of true three body scattering in HOR. As an illustration the soft dipole mode in the (11)Li nucleus is considered in a frame of the (9)Li+n+n cluster model taking into account of three body continuum effects.

"On an Application of the New Atomic Theory to the Graphical Representation of the Invariants and Covariants of Binary Quantics, With Three Appendices, [Continued]" is an article from American Journal of Mathematics, Volume 1 . View more articles from American Journal of Mathematics . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-2369302

In this paper we give a combinatorial characterization of projections of geodesics in Euclidean buildings to Weyl chambers. We apply these results to the representation theory of complex semisimple Lie groups and to spherical Hecke rings associated with nonarchimedean reductive Lie groups. Our main application is a generalization of the saturation theorem of Knutson and Tao for SL(n) to other complex semisimple Lie groups.

Book digitized by Google from the library of the University of Michigan and uploaded to the Internet Archive by user tpb.

Book digitized by Google from the library of the University of Michigan and uploaded to the Internet Archive by user tpb.

Book digitized by Google from the library of the University of Michigan and uploaded to the Internet Archive by user tpb.

The observed probabilities of quantum mechanics possess a time asymmetry which is based on the truism that a state must be prepared before an observable can be measured in it. While Hilbert space quantum theory cannot incorporate this arrow of time, the Rigged Hilbert Space (RHS) formulation of quantum mechanics provides a theory of time symmetric as well as time asymmetric quantum physics.

We call one-way simple random walk a random walk in the quadrant Z_+^n whose increments belong to the canonical base. In relation with representation theory of Lie algebras and superalgebras, we describe the law of such a random walk conditioned to stay in a closed octant, a semi-open octant or other types of semi-groups. The combinatorial representation theory of these algebras allows us to describe a generalized Pitman transformation which realizes the conditioning on the set of paths of the walk. We pursue here in a direction initiated by O'Connell and his coauthors [13,14,2], and also developed in [12]. Our work relies on crystal bases theory and insertion schemes on tableaux described by Kashiwara and his coauthors in [1] and, very recently, in [5].

In 1998 the Adapted Ordering Method was developed for the study of the representation theory of the superconformal algebras in two dimensions. It allows: to determine the maximal dimension for a given type of space of singular vectors, to identify all singular vectors by only a few coefficients, to spot subsingular vectors and to set the basis for constructing embedding diagrams. In this talk I introduce the present version of the Adapted Ordering Method, published in J. Phys. A: Math. Theor. 41 (2008) 045201, which can be applied to general Lie algebras and superalgebras and their generalizations, provided they can be triangulated.

We develop a new approach to the representation theory of quantum algebras supporting a torus action via methods from the theory of finite-state automata and algebraic combinatorics. We show that for a fixed number $m$, the torus-invariant primitive ideals in $m\times n$ quantum matrices can be seen as a regular language in a natural way. Using this description and a semigroup approach to the set of Cauchon diagrams, a combinatorial object that paramaterizes the primes that are torus-invariant, we show that for $m$ fixed, the number of torus-invariant primitive ideals in $m\times n$ quantum matrices satisfies a linear recurrence in $n$ over the rational numbers. In the $3\times n$ case we give a concrete description of the torus-invariant primitive ideals and use this description to give an explicit formula for the number P(3,n).

We study some distributive lattices arising in the combinatorics of lattice paths. In particular, for the Dyck, Motzkin and Schroder lattices we describe the spectrum and we determine explicitly the Euler characteristic in terms of natural parameters of lattice paths.

We study some distributive lattices arising in the combinatorics of lattice paths. In particular, for the Dyck, Motzkin and Schroder lattices we describe the spectrum and we determine explicitly the Euler characteristic in terms of natural parameters of lattice paths.

In this paper, by the use of Potential Theory, some representation results for multivariate functions from the Sobolev spaces in terms of the double layer potential and the fundamental solution of Laplace's equation are pointed out. Applications for multivariate inequalities of Ostrowski type are also provided.

In this paper we study the finitely generated algebras underlying $W$ algebras. These so called 'finite $W$ algebras' are constructed as Poisson reductions of Kirillov Poisson structures on simple Lie algebras. The inequivalent reductions are labeled by the inequivalent embeddings of $sl_2$ into the simple Lie algebra in question. For arbitrary embeddings a coordinate free formula for the reduced Poisson structure is derived. We also prove that any finite $W$ algebra can be embedded into the Kirillov Poisson algebra of a (semi)simple Lie algebra (generalized Miura map). Furthermore it is shown that generalized finite Toda systems are reductions of a system describing a free particle moving on a group manifold and that they have finite $W$ symmetry. In the second part we BRST quantize the finite $W$ algebras. The BRST cohomology is calculated using a spectral sequence (which is different from the one used by Feigin and Frenkel). This allows us to quantize all finite $W$ algebras in one stroke. Explicit results for $sl_3$ and $sl_4$ are given. In the last part of the paper we study the representation theory of finite $W$ algebras. It is shown, using a quantum version of the generalized Miura transformation, that the representations of finite $W$ algebras can be constructed from the representations of a certain Lie subalgebra of the original simple Lie algebra. As a byproduct of this we are able to construct the Fock realizations of arbitrary finite $W$ algebras.

We consider separately radial (with corresponding group $\mathbb{T}^n$) and radial (with corresponding group $\mathrm{U}(n))$ symbols on the projective space $\mathbb{P}^n(\mathbb{C})$, as well as the associated Toeplitz operators on the weighted Bergman spaces. It is known that the $C^*$-algebras generated by each family of such Toeplitz operators are commutative. We present a new representation theoretic proof of such commutativity. Our method is easier and more enlightening as it shows that the commutativity of the $C^*$-algebras is a consequence of the existence of multiplicity-free representations. Furthermore, our method shows how to extend the current formulas for the spectra of the corresponding Toeplitz operators to any closed group lying between $\mathbb{T}^n$ and $\mathrm{U}(n)$.

A summary of the construction procedure of generalized versions of Baxter's Q-operator is given. Illustrated by several figures and diagrams the use of representation theory is explained step-by-step avoiding technical details. The relation with the infinite-dimensional non-abelian symmetries of the six-vertex model at roots of unity is discussed and parallels with the eight-vertex case outlined.

We construct a canonical irreducible representation for the orthofermion algebra of arbitrary order, and show that every representation decomposes into irreducible representations that are isomorphic to either the canonical representation or the trivial representation. We use these results to show that every orthosupersymmetric system of order $p$ has a parasupersymmetry of order $p$ and a fractional supersymmetry of order $p+1$.

We construct a canonical irreducible representation for the orthofermion algebra of arbitrary order, and show that every representation decomposes into irreducible representations that are isomorphic to either the canonical representation or the trivial representation. We use these results to show that every orthosupersymmetric system of order $p$ has a parasupersymmetry of order $p$ and a fractional supersymmetry of order $p+1$.

Bousfield recently gave a formula for the odd-primary v1-periodic homotopy groups of a finite H-space in terms of its K-theory and Adams operations. We apply Bousfield's theorem to give explicit determinations of the v1-periodic homotopy groups of (E8,5) and (E8,3), thus completing the determination of all odd-primary v1-periodic homotopy groups of compact simple Lie groups, a project suggested by Mimura in 1989. The method involves no homotopy theoretic input and no spectral sequences. The input is the second eexterior power operation in the representation ring of E8, which we determine using specialized software. This can be interpreted as giving the Adams operation psi^2 in K(E8). Eigenvectors of psi^2 must also be eigenvectors of psi^k for any k. The matrix of these eigenvectors is the key to the analysis. Its determinant is closely related to the homotopy decomposition of E8 localized at each prime. By taking careful combinations of eigenvectors, we obtain a set of generators of K(E8) on which we have a nice formula for all Adams operations. Bousfield's theorem (and much Maple computation) allows us to obtain the v1-periodic homotopy groups.

Book digitized by Google from the library of University of California and uploaded to the Internet Archive by user tpb.