Structure of the unitary valuation algebra

By Joseph H. G. Fu. 2004

84 citations recorded by the source corpus.

S. Alesker has shown that if $G$ is a compact subgroup of O(n) acting transitively on the unit sphere $S^{n-1}$ then the vector space $Val^G$ of continuous, translation-invariant, $G$-invariant convex valuations on $R^n$ has the structure of a finite dimensional graded algebra over $R$ satisfying Poincare duality. We show that the kinematic formulas for $G$ are determined by the product pairing. Using this result we then show that the algebra $Val^{U(n) }$ is isomorphic to $R[s,t]/(f_{n+1}, f_{n+2})$, where $s,t$ have degrees 2 and 1 respectively, and the polynomial $f_i$ is the degree $i$ term of the power series $\log(1 + s +t)$.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0757ea8603052fd5678e3336bf242eb39502e5c6

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Theory of Valuations on Manifolds: A Survey (2006, background): “Fu [25] in terms of generators and relations.”
Harmonic Analysis of Translation Invariant Valuations (2010, background): “For generalizations of this idea in different directions, we refer to [3, 5, 12, 16, 18, 31, 37, 47].”
A classification of $\operatorname{SL}(n)$ invariant valuations (2010, background): “Since then investigations of valuations have been an active and prominent part of mathematics (see [1]–[8], [16], [19], [20], [24], [25], [31]–[34], and [53] for some of the more recent results).”
Integral geometry of complex space forms (2012, background): “If λ = 0 then this definition of s is identical to that in [32, 19].”
Fisher information and matrix-valued valuations (2011, result): “See [24, 27] for information on the classical theory of valuations on convex sets and [1–5,9,11,16,19–22,28– 31, 33, 34, 43–45] for some of the more recent results.”
Algebraic Integral Geometry (2010, background): “For this, Fu [34] defined the kinematic operator kG : Val G → Val ⊗Val”
The product on smooth and generalized valuations (2009, unknown): “(ϕs,ϕt) = X cµ i,jhpGϕi,φsihpGϕj,ϕti = X gikgjlg isgjthpGµ,ϕk · ϕli = hpGµ,ϕs ·ϕti = m∗ G ◦ pG(µ)(ϕs,ϕti. Remarks: (1) A similar theorem in the translation invariant case was proven in [17], based on [24]. Roughly speaking, the algebra structure on V∞(X)G (respectively ValG in the translation invari”
Minkowski valuations and generalized valuations (2015, background): “These results in turn were critical for the tremendous progress in integral geometry of recent years (see [5, 16, 17, 20, 35, 66] and the references therein).”
Valuations on Sobolev spaces (2012, background): “See [25, 30, 47, 48] for information on the classical theory of valuations on convex bodies and [1, 2, 3, 4, 5, 9, 16, 21, 20, 22, 23, 33, 34, 35, 38, 39, 51, 53, 54, 58] for some of the more recent results.”
Projection bodies in complex vector spaces (2011, background): “The classical case A = R has attracted a lot of attention, we refer to [1, 2, 3, 4, 7, 8, 13, 20, 29] for some new developments.”
Difference bodies in complex vector spaces (2012, background): “For some references on this direction see [4, 10, 12, 13, 14].”
Minkowski Areas and Valuations (2010, background): “of elementary mixed volumes, valuations have become a critical notion (see [27, 31, 47] and see [1, 2, 3, 4, 5, 6, 11, 15, 29, 30, 49, 51, 52, 53] for some of the more recent contributions).”
Crofton Measures and Minkowski Valuations (2010, unknown): “, [1–5, 12, 14, 20, 25, 28, 30, 33, 34].”
Valuations and Surface Area Measures (2014, unknown): “[2–6, 9, 13–15, 20–25, 34, 39–41]).”
$\mathrm{GL}(n)$ contravariant Minkowski valuations (2012, unknown): “, [1–5, 7, 15–17, 19, 22].”