Schur Partial Derivative Operators

A lattice diagram is a finite list L=((p_1,q_1),...,(p_n,q_n) of lattice cells. The corresponding lattice diagram determinant is \Delta_L(X;Y)=\det \| x_i^{p_j}y_i^{q_j} \|. These lattice diagram determinants are crucial in the study of the so-called ``n! conjecture'' of A. Garsia and M. Haiman. The space M_L is the space spanned by all partial derivatives of \Delta_L(X;Y). The ``shift operators'', which are particular partial symmetric derivative operators are very useful in the comprehension of the structure of the M_L spaces. We describe here how a Schur function partial derivative operator acts on lattice diagrams with distinct cells in the positive quadrant.Comment: 8 pages, LaTe

Combinatorics of Labelled Parallelogram polyominoes

We obtain explicit formulas for the enumeration of labelled parallelogram polyominoes. These are the polyominoes that are bounded, above and below, by north-east lattice paths going from the origin to a point (k,n). The numbers from 1 and n (the labels) are bijectively attached to the $n$ north steps of the above-bounding path, with the condition that they appear in increasing values along consecutive north steps. We calculate the Frobenius characteristic of the action of the symmetric group S_n on these labels. All these enumeration results are refined to take into account the area of these polyominoes. We make a connection between our enumeration results and the theory of operators for which the intergral Macdonald polynomials are joint eigenfunctions. We also explain how these same polyominoes can be used to explicitly construct a linear basis of a ring of SL_2-invariants.Comment: 25 pages, 9 figure

Ideals of Quasi-Symmetric Functions and Super-Covariant Polynomials for S_n

The aim of this work is to study the quotient ring R_n of the ring Q[x_1,...,x_n] over the ideal J_n generated by non-constant homogeneous quasi-symmetric functions. We prove here that the dimension of R_n is given by C_n, the n-th Catalan number. This is also the dimension of the space SH_n of super-covariant polynomials, that is defined as the orthogonal complement of J_n with respect to a given scalar product. We construct a basis for R_n whose elements are naturally indexed by Dyck paths. This allows us to understand the Hilbert series of SH_n in terms of number of Dyck paths with a given number of factors.Comment: LaTeX, 3 figures, 12 page

On the symmetry of the partition function of some square ice models

We consider the partition function Z(N;x_1,...,x_N,y_1,...,y_N) of the square ice model with domain wall boundary. We give a simple proof of the symmetry of Z with respect to all its variables when the global parameter a of the model is set to the special value a=exp(i\pi/3). Our proof does not use any determinantal interpretation of Z and can be adapted to other situations (for examples to some symmetric ice models).Comment: 8 page

Evolution of microstructure and crystallographic texture during dissimilar friction stir welding of duplex stainless steel to low carbon-manganese structural steel

Electron backscattered diffraction (EBSD) was used to analyze the evolution of microstructure and crystallographic texture during friction stir welding of dissimilar type 2205 duplex stainless steel (DSS) to type S275 low carbon-manganese structural steel. The results of microstructural analyses show that the temperature in the center of stirred zone reached temperatures between Ac 1 and Ac 3 during welding, resulting in a minor ferrite-to-austenite phase transformation in the S275 steel, and no changes in the fractions of ferrite and austenite in the DSS. Temperatures in the thermomechanically affected and shoulder-affected zones of both materials, in particular toward the root of the weld, did not exceed the Ac 1 of S275 steel. The shear generated by the friction between the material and the rotating probe occurred in austenitic/ferritic phase field of the S275 and DSS. In the former, the transformed austenite regions of the microstructure were transformed to acicular ferrite, on cooling, while the dual-phase austenitic/ferritic structure of the latter was retained. Studying the development of crystallographic textures with regard to shear flow lines generated by the probe tool showed the dominance of simple shear components across the whole weld in both materials. The ferrite texture in S275 steel was dominated by D 1, D 2, E, E¯ , and F, where the fraction of acicular ferrite formed on cooling showed a negligible deviation from the texture for the ideal shear texture components of bcc metals. The ferrite texture in DSS was dominated by D 1, D 2, I, I¯ , and F, and that of austenite was dominated by the A, A¯ , B, and B¯ of the ideal shear texture components for bcc and fcc metals, respectively. While D 1, D 2, and F components of the ideal shear texture are common between the ferrite in S275 steel and that of dual-phase DSS, the preferential partitioning of strain into the ferrite phase of DSS led to the development of I and I¯ components in DSS, as opposed to E and E¯ in the S275 steel. The formations of fine and ultrafine equiaxed grains were observed in different regions of both materials that are believed to be due to strain-induced continuous dynamic recrystallization (CDRX) in ferrite of both DSS and S275 steel, and discontinuous dynamic recrystallization (DDRX) in austenite phase of DSS