Comultiplication rules for the double Schur functions and Cauchy identities

The double Schur functions form a distinguished basis of the ring \Lambda(x||a) which is a multiparameter generalization of the ring of symmetric functions \Lambda(x). The canonical comultiplication on \Lambda(x) is extended to \Lambda(x||a) in a natural way so that the double power sums symmetric functions are primitive elements. We calculate the dual Littlewood-Richardson coefficients in two different ways thus providing comultiplication rules for the double Schur functions. We also prove multiparameter analogues of the Cauchy identity. A new family of Schur type functions plays the role of a dual object in the identities. We describe some properties of these dual Schur functions including a combinatorial presentation and an expansion formula in terms of the ordinary Schur functions. The dual Littlewood-Richardson coefficients provide a multiplication rule for the dual Schur functions.Comment: 44 pages, some corrections are made in sections 2.3 and 5.

Tyler shape depth

In many problems from multivariate analysis, the parameter of interest is a shape matrix, that is, a normalized version of the corresponding scatter or dispersion matrix. In this paper, we propose a depth concept for shape matrices that involves data points only through their directions from the center of the distribution. We use the terminology Tyler shape depth since the resulting estimator of shape, namely the deepest shape matrix, is the median-based counterpart of the M-estimator of shape of Tyler (1987). Beyond estimation, shape depth, like its Tyler antecedent, also allows hypothesis testing on shape. Its main benefit, however, lies in the ranking of shape matrices it provides, whose practical relevance is illustrated in principal component analysis and in shape-based outlier detection. We study the invariance, quasi-concavity and continuity properties of Tyler shape depth, the topological and boundedness properties of the corresponding depth regions, existence of a deepest shape matrix and prove Fisher consistency in the elliptical case. Finally, we derive a Glivenko-Cantelli-type result and establish almost sure consistency of the deepest shape matrix estimator.Comment: 28 pages, 5 figure

Ninth variation of classical group characters of type A-D and Littlewood identities

We introduce certain generalisations of the characters of the classical Lie groups, extending the recently defined factorial characters of Foley and King. This is done by replacing the factorial powers with a sequence of polynomials. In particular, we offer a ninth variation generalisation for the rational Schur functions. We derive Littlewood-type identities for our generalisations. These identities allow us to give new unflagged Jacobi-Trudi identities for the factorial characters of types A-D. We also propose an extension of the original Macdonald's ninth variation to the case of symplectic and orthogonal characters, which helps us prove N\"agelsbach-Kostka identities.Comment: 29 pages; double dual Schur functions redefined, ninth variation type A Littlewood identity restated, Jacobi-Trudi and N\"agelsbach-Kostka identities added, another ninth variation introduced for the types CB