Let V be a finite dimensional real vector space. Let V alsm(V ) be the space of translation invariant smooth valuations on convex compact subsets of V . Let Dens(V ) be the space of Lebesgue measures on V . The goal of the article is to construct and study an isomorphism FV : V alsm(V )−̃→V alsm(V ∗)⊗Dens(V ) such that FV commutes with the natural action of the full linear group on both spaces, ...

Consider random Young diagrams with fixed number n of boxes, distributed according to the Plancherel measure Mn. That is, the weight Mn(λ) of a diagram λ equals dim λ/n!, where dimλ denotes the dimension of the irreducible representation of the symmetric group Sn indexed by λ. As n → ∞, the boundary of the (appropriately rescaled) random shape λ concentrates near a curve Ω (Logan– Shepp 1977, V...

ABSTRACT. We study the shape of the Young diagram λ associated via the Robinson–Schensted– Knuth algorithm to a random permutation in Sn such that the length of the longest decreasing subsequence is not bigger than a fixed number d; in other words we study the restriction of the Plancherel measure to Young diagrams with at most d rows. We prove that in the limit n → ∞ the rows of λ behave like ...

The aim of these two papers (I and II) is to try to give fundamental concepts of quantum kinematics to q-deformed quantum spaces. Paper I introduces the relevant mathematical concepts. A short review of the basic ideas of q-deformed analysis is given. These considerations are continued by introducing q-deformed analogs of Fourier transformations and delta functions. Their properties are discuss...