1 the electronic journal of combinatorics 5 (1998), #R38 2 Multiple zeta values (MZVs, also called Euler sums or multiple harmonic series) are nested generalizations of the classical Riemann zeta function evaluated at integer values. The fact that an integral representation of MZVs obeys a shuffle product rule allows the possibility of a combinatorial approach to them. Using this approach we prove a longstanding conjecture of Don Zagier about MZVs with certain repeated arguments. We also prove a similar cyclic sum identity. Finally, we present extensive computational evidence supporting an infinite family of conjectured MZV identities that simultaneously generalize the Zagier identity. 1

Abstract. For a positive definite fundamental tensor all known examples of Osserman algebraic curvature tensors have a typical structure. They can be produced from a metric tensor and a finite set of skew-symmetric matrices which fulfil Clifford commutation relations. We show by means of Young symmetrizers

We prove an elegant combinatorial rule for the generation of Schubert polynomials based on box diagrams, which was conjectured by A. Kohnert. The main tools for the proof are (1) a recursive structure of Schubert polynomials and (2) a partial order on the set of box diagrams. As a byproduct we obtain (combinatorial) proofs for two other rules for the generation of Schubert polynomials based on box diagrams: (1) the more complicated rule of N. Bergeron, and (2) the rule of P. Magyar, which we show to be a simplified Bergeron rule.

Introduction and results In the present paper we apply the theory of skew-symmetric vanishing lattices developed around 15 years ago by B. Wajnryb, S. Chmutov, and W. Janssen for the necessities of the singularity theory to the enumeration of connected components in the intersection of two open opposite Schubert cells in the space of complete real flags. Let us briefly recall the main topological problem considered in [SSV] and reduced there to a group-theoretical question solved below. Let N be the group of real unipotent uppertriangular (n+ 1) \Theta (n + 1) matrices and D i be the determinant of the submatrix formed by the first i rows and the last i columns. Denote by \Delta i the divisor fD i = 0g ae N and let \Delta be the union [ i=1 \Delta i . Consider now the complement U = N n \Delta . The space U can be interpreted as the intersection of two open opposite Schubert cells in SLn+1 (R)=B. In [SSV] we have studied the number of connected components in U

this paper we give a K-linear isomorphism ! of the free associative algebra A of countable rank onto \Delta such that (i) fl (A) is mapped onto (Rad \Delta) for all j 0, (ii) the multihomogeneous components of A are mapped onto a direct decomposition of \Delta into indecomposable left ideals

We use the Hopf algebra structure of the algebra of symmetric functions to study the Adams operators of the complex representation rings of symmetric groups, and we give new proofs of all of Littlewood's formulas for inner plethysm. We also study the Adams operations for orthogonal and symplectic group characters. 1 Background Our notations for symmetric functions (of an infinite, countable set of indeterminates X = fx 1 ; x 2 ; : : :g) will be the same as in [Mcd], except for the following minor changes. We denote by Sym(X) := n Sym (X) the Z-algebra of symmetric functions over X , graded by total degree. The standard scalar product on Sym(X) (for which the Schur functions form an orthonormal basis) is denoted by (\Delta ; \Delta) rather than by h\Delta ; \Deltai, the symbol h\Deltai being reserved for an other use. Also, for a symmetric function F , we denote by D F instead of D(F ) the adjoint of the linear operator G 7! FG. The generating series for complete and elementary functions are oe z (X) = h n (X) = (1 \Gamma zx i ) z (X) = e n (X) = (1 + zx i ) : A partition can be described by the sequence of its parts, arranged in decreasing order or by the multiplicities of the parts. If ff = (ff 1 ; ff 2 ; : : :) = (1 : : :), then we call the sequence a := (a 1 ; a 2 ; : : :) the cycle type of ff.

Abstract. Symmetry properties of r-times covariant tensors T can be described by certain linear subspaces W of the group ring K[Sr] of a symmetric group Sr. If for a class of tensors T such a W is known, the elements of the orthogonal subspace W ⊥ of W within the dual space K[Sr] ∗ of K[Sr] yield linear identities needed for a treatment of the term combination problem for the coordinates of the T. We give the structure of these W for every situation which appears in symbolic tensor calculations by computer. Characterizing idempotents of such W can be determined by means of an ideal decomposition algorithm which works in every semisimple ring up to an isomorphism. Furthermore, we use tools such as the Littlewood-Richardson rule, plethysms and discrete Fourier transforms for Sr to increase the efficience of calculations. All described methods were implemented in a Mathematica package called PERMS. 1. The Term Combination Problem for Tensors The use of computer algebra systems for symbolic calculations with tensor expressions is very important in differential geometry, tensor analysis and general relativity theory. The investigations of this paper1 are motivated by the following term combination problem or normal form problem which occurs within such calculations. Let us consider real or complex linear combinations n∑ (1.1) τ = , αi ∈ R, C i=1 αiT(i) of expressions T(i) which are formed from the coordinates of certain tensors A, B, C,... by multiplication and, possibly, contractions of some pairs of indices. An example of such an expression is (1.2) A iabc A a jkd Bbd e Cec. In (1.2) we use Einstein’s summation convention. Further we assume that each of the numbers of A, B, C,... is constant if we run through the set of the T(i). Now we aim to carry out symbolic calculations with expressions of the type (1.1), (1.2)

this paper we obtain a description of the descending Loewy series ((Rad \Delta n ) ) j0 of \Delta n

Abstract. We show that the space of algebraic covariant derivative curvature tensors R ′ is generated by Young symmetrized product tensors T ⊗ ˆ T or ˆ T ⊗ T, where T and ˆ T are covariant tensors of order 2 and 3 whose symmetry classes are irreducible and characterized by the following pairs of partitions: {(2), (3)}, {(2), (2 1)} or {(1 2), (2 1)}. Each of the partitions (2), (3) and (1 2) describes exactly one symmetry class, whereas the partition (2 1) characterizes an infinite set S of irreducible symmetry classes. This set S contains exactly one symmetry class S0 ∈ S whose elements ˆ T ∈ S0 can not play the role of generators of tensors R ′. The tensors ˆ T of all other symmetry classes from S \ {S0} can be used as generators for tensors R ′. Foundation of our investigations is a theorem of S. A. Fulling, R. C. King, B. G. Wybourne and C. J. Cummins about a Young symmetrizer that generates the symmetry class of algebraic covariant derivative curvature tensors. Furthermore we apply ideals and idempotents in group rings C[Sr], the Littlewood-Richardson rule and discrete Fourier transforms for symmetric groups Sr. For certain symbolic calculations we used the Mathematica packages Ricci and PERMS. 1.

In the constructive theory of linear codes, we can restrict attention to the isometry classes of indecomposable codes, as it was shown by Slepian.