A Hadwiger-Type Theorem for the Special Unitary Group

By A. Bernig. 2008

67 citations recorded by the source corpus.

The dimension of the space of SU(n) and translation-invariant continuous valuations on $${\mathbb {C}^n}$$, n ≥ 2, is computed. For even n, this dimension equals (n2 + 3n + 10)/2; for odd n it equals (n2 + 3n + 6)/2. An explicit geometric basis of this space is constructed. The kinematic formulas for SU(n) are obtained as corollaries.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1cf779abe7c344b1f09f6a6a2ae1c79b0df57532

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. A. Bernig
2. Thomas Wannerer
3. Franz E. Schuster
4. Monika Ludwig
5. Christoph Haberl
6. Lukas Parapatits
7. Yiming Zhao
8. Semyon Alesker

In Geometry and Topology

1. A. Bernig
2. Franz E. Schuster
3. Thomas Wannerer
4. Joseph H. G. Fu
5. Semyon Alesker
6. Christoph Haberl
7. Monika Ludwig
8. Lukas Parapatits

In Mathematical Physics

1. A. Bernig
2. Franz E. Schuster
3. Thomas Wannerer
4. Christoph Haberl
5. Joseph H. G. Fu
6. Lukas Parapatits
7. G. Solanes
8. Semyon Alesker

Citations in the corpus

• Algebraic Integral Geometry (2010, background): “The algebra structure and the kinematic formulas for SU(n) are variations from the U(n)-case, see [22].”
• Valuations and Surface Area Measures (2014, unknown): “[2–6, 9, 13–15, 20–25, 34, 39–41]).”
• Integral geometry of complex space forms (2012, background): “Using this approach, the integral geometry of the spaces (V,H) = (Cn,U(n)), (Cn, SU(n)), (R7,G2), (R8, Spin(7)) has been successfully analyzed in [13, 14, 15, 19].”
• The module of unitarily invariant area measures (2012, background): “The restrictions of these forms to the sphere bundle generate the algebra of U(n)-invariant forms on SC, see [21] or, for the case of SU(n)-invariant forms, [16].”
• The Centro-Affine Hadwiger Theorem (2013, background): “ long conjectured property, finally established in [48]) was crucial for the solution of the affine Plateau problem [72]. Hadwiger type theorems were established also in various other contexts (see e.g. [10,32,33,35, 43,44]). Of particular importance are classification theorems of body valued valuations”
• The product on smooth and generalized valuations (2009, background): “tions. A similar theorem in the affine, translation invariant case was proved in [17]. It was the starting point to write down in an explicit form all kinematic formulas for the groups U(n) [18], SU(n) [14], G2 and Spin(7) [15]. Remark 1.1. Compact rank one symmetric spaces have been classified by E. C”
• 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].”
• Minkowski valuations and generalized valuations (2015, background): “, [4, 9, 13, 31, 43]) and, more general, equivariant tensor valued valuations, where A = SymRn (see [3, 10, 35, 67]).”
• Minkowski valuations intertwining the special linear group (2012, background): “, [1–4, 8, 11, 15, 20, 33, 36, 40, 41]), the study of valuations headed mainly in two different directions during the last years.”