Representations and Invariants of the Classical Groups

By R. Goodman, Nolan Wallach. 1998

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Citations in the corpus

Gelfand-Tsetlin bases for classical Lie algebras (2002, methodology): “The reader is refered to, e.g., the books of Bourbaki [11], Dixmier [19], Humphreys [62] or Goodman and Wallach [45] for a detailed exposition of the theory.”
Classical Algebraic Geometry: A Modern View (2012, background): “The linear representationH q (E) is an irreducible representation of O(E, q) (see [253]).”
STRUCTURED TOOLS FOR STRUCTURED MATRICES (2003, background): “The matrices G which preserve the value of the scalar product Gx, Gy = x, y, ∀x, y ∈ K n .”
On the ideals of equivariant tree models (2007, background): “The general Markov model associated to T is the algebraic variety where the closure is taking in the Zariski topology.”
Optimal sequence of quantum measurements in the sense of Stein's lemma in quantum hypothesis testing (2002, background): “Concerning the natural representation πSL(H) of the special linear group SL(H) on H, we consider its n-th tensored representation π⊗n SL(H) := πSL(H) ⊗ · · · ⊗ πSL(H) } {{ } n on the tensored space H⊗n [21, 22].”
Hairy graphs and the unstable homology of Mod(g, s), Out(Fn) and Aut(Fn) (2011, background): “ an isomorphism, we may instead compute H1(GL 2(Z);C[V V]): Set P V = C[V V] = C[V C2]. Then P V is a GL(V) GL 2(C)-module, which by Schur-Weyl duality can be decomposed as P V = M S V S C2: (See [9] p.218 and p.257.) The Weyl module S C2 is zero unless = (k;l) is a Young diagram with only two rows”
The Church of the Symmetric Subspace (2013, background): “[18, 33] are excellent, but approach the subject respectively from the Lie-algebraic or combinatorial perspective, rather than in terms of quantum information.”
Howe Duality for Lie Superalgebras (2000, methodology): “s of shapes (k1,k1+1),... ,(ks,ks+1) with ki > ki+1 for i = 1,... ,s−1 is done in an analogous fashion. Now we can state the following multiplicity-free decompositions of gl(m)- modules (cf. [H2], [GW]): Λ(S2Cm) ∼= X λ Vλ (2.5) m, Λ(Λ2Cm) ∼= X µ Vµ (2.6) m, where λ (respectively µ) is over all pa”
Schur-Weyl duality and the heat kernel measure on the unitary group (2007, background): “The correct statement of Schur-Weyl duality in the present context is that the subalgebras of End((C )⊗n) generated by SO(N) and Bn are each other’s commutant (see [11]).”
Tapering off qubits to simulate fermionic Hamiltonians (2017, background): “It is well-known [25, 26] that Πλ,T is proportional to a (non-orthogonal) projector onto a subspace of the sector λ in the Schur decomposition.”
Random Quantum Circuits are Approximate 2-designs (2008, background): “[15] and for a less precise proof but from a quantum information perspective see Ref.”
Harmonic Analysis of Translation Invariant Valuations (2010, methodology): “As a general reference for this material we recommend the books by Bröcker and tom Dieck [17], Fulton and Harris [19], and Goodman and Wallach [21].”
On spherical harmonics for fuzzy spheres in diverse dimensions (2001, background): “matrix algebra. This is Burnside’s theorem, described for example in [ 27 ].”
Stable branching rules for classical symmetric pairs (2003, methodology): “In each of these cases, we select a Borel subalgebra of the classical Lie algebra as is done in [GW].”