geometric representation theory
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Be?linson-Bernstein localization?
The Schur-Weyl measure is probability distribution on the set of Young diagrams of given number of boxes, closely related to expressions appearing in Schur-Weyl duality.
For , write:
for the set of partitions of , hence of Young diagrams with boxes;
for write:
for the set of standard Young tableaux of shape ;
for write
Then the Schur-Weyl measure for is
(e.g. Petrov 19, slide 76)
In the special case that , and writing
for the complex irreducible representation of the symmetric group which is labeled by (the Specht module, see the representation theory of the symmetric group);
for the complex irreducible representation of the (special) general linear group which is labeled by (see the representation theory of the general linear group),
this is equal to (see at hook length formula):
With respect to the Schur-Weyl measure on and in the limit of large , the logarithm of the Schur-Weyl probability is almost surely approximately constant (i.e. independent of ) on the value
for some , in that for all we have
The Schur-Weyl measure is the pushforward measure of the probability distribution on pure states encoded by the “Cayley state” (here): the quantum state that is given by the Cayley distance kernel – see there for details.
Review:
Leonid Petrov, slides 73-76 in: Random Matrices, lecture notes 2019 (pdf slides, pdf, webpage)
(in the context of random matrix theory)
See also:
Sevak Mkrtchyan, Entropy of Schur-Weyl Measures, Annales de l’I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 2, pp. 678-713. (arXiv:1107.1541, numdam:AIHPB_2014__50_2_678_0)
Pierre-Loïc Méliot, Kerov’s central limit theorem for Schur-Weyl and Gelfand measures, DMTCS proc. AO, 2011, 669–680 (pdf)
Pierre-Loïc Méliot, Kerov’s central limit theorem for Schur-Weyl measures of parameter 1/2 (arXiv:1009.4034)
M. S. Boyko and N. I. Nessonov, Entropy of the Shift on -representations of the Group (pdf)
Last revised on June 3, 2021 at 19:14:48. See the history of this page for a list of all contributions to it.