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Unfortunately, I need to discuss with you another terminological problem. I am lightly doing a circle of entries related to combinatorial aspects of representation theory. I stumbled accross permutation representation entry. It says that the permutation representation is the representation in category Set. Well, nice but not that standard among representation theorists themselves. Over there one takes such a thing – representation by permutations of a finite group G on a set X, and looks what happens in the vector space of functions into a field K. As we know, for a group element g the definition is, (gf)(x)=f(g−1x), for f:X→K is the way to induce a representation on the function space KX. The latter representation is called the permutation representation in the standard representation theory books like in
I know what to do approximately, we should probably keep both notions in the entry (and be careful when refering to this page – do we mean representation by permutations, what is current content or permutation representation in the rep. theory on vector spaces sense). But maybe people (Todd?) have some experience with this terminology.
Edit: new (related) entries for Claudio Procesi and Arun Ram.
Added:
There are two inequivalent definitions of Fréchet spaces found in the literature. The original definition due to Stefan Banach defines Fréchet spaces as metrizable complete topological vector spaces.
Later Bourbaki (Topological vector spaces, Section II.4.1) added the condition of local convexity. However, many authors continue to use the original definition due to Banach.
The term “F-space” can refer to either of these definitions, although in the modern literature it is more commonly used to refer to the non-locally convex notion.
The nLab uses “F-space” to refer to the non-locally convex notion and “Fréchet space” to refer to the locally convex notion.
I added a Definition section to Burnside ring (and made Burnside rig redirect to it).
stub for quantum computation
added at adjoint functor
more details in the section In terms of universal arrows;
a bit in the section Examples
Added to Dedekind cut a short remark on the ¬¬-stability of membership in the lower resp. the upper set of a Dedekind cut.
I have added some accompanying text to the list of links at monad (disambiguation).
One question: in the entry Gottfried Leibniz it is claimed that the term “monad” for a functor on a category with monoid structure also follows Leibniz’s notion of monads. Is this really so? What’s a reference for this claim?
I am asking because I don’t see how the notion of monoid in the endomorphisms of a category would be related to what Leibniz was talking about. What’s the idea, if there is one?
added reference to derived category
added pointer to:
Julian Schwinger: Quantum Kinematics and Dynamics, CRC Press (1969, 1991) [ISBN:9780738203034, pdf]
Julian Schwinger (ed.: Berthold-Georg Englert): Quantum Mechanics – Symbolism of Atomic Measurements, Springer (2001) [doi:10.1007/978-3-662-04589-3]
with a link to arguments that Schwinger secretly (re-)invented groupoid algebra, in these books.
a bare list of references on arguments
(by Connes) that Heisenberg’s original derivation of “matrix mechanics” and
more generally (by Ibort et al.) that Schwinger’s less known “algebra of selective measurements”
are both best understood, in modern language, as groupoid convolution algebras,
to be !include-ed into relevant entries (such as quantum observables and groupoid algebra), for ease of synchronizing
a stub entry, for the moment just to make a link work that has long been requested at Handbook of Quantum Gravity
for the equivariant+twisted version I added further pointer to
El-kaïoum M. Moutuou, Graded Brauer groups of a groupoid with involution, J. Funct. Anal. 266 (2014), no.5 (arXiv:1202.2057)
Daniel Freed, Gregory Moore, Section 7 of: Twisted equivariant matter, Ann. Henri Poincaré (2013) 14: 1927 (arXiv:1208.5055)
Kiyonori Gomi, Freed-Moore K-theory (arXiv:1705.09134, spire:1601772)
just the other day I was searching for good references on “asymptotic symmetries”, not finding much. But today appears the useful
and so I am starting an entry hereby
a bare list of references, to be !include-ed into the References-sections of relevant entries (such as knot homology, topological string theory) for ease of synchronization
added at Grothendieck universe at References a pointer to the proof that these are sets of κ-small sets for inaccessible κ. (also at inaccessible cardinal)
Together with my PhD students, I have been thinking a lot recently about the appropriate notion of a module over a C^∞-ring, i.e., something with better properties than Beck modules, which boil down to modules over the underlying real algebra in this case.
We stumbled upon the article C-infinity module (schreiber).
It says: “a C-infty algebra A is a copresheaf A∈Quantities=CoPrSh(CartesianSpaces) which becomes a copresheaf with values in algebras when restricted along FinSet↪CartesianSpaces,”
Why are we restricting to FinSet here? The underlying commutative real algebra is extracted by restricting to the Lawvere theory of commutative real algebras, i.e., CartesianSpaces_Poly, the subcategory of cartesian spaces and polynomial maps. Restricting to FinSet^op (as opposed to FinSet) extracts the underlying set only. It is unclear what is being meant by restricting along FinSet→CartesianSpaces, since the latter functor does not preserve finite products, so restricting along it does not produce a functor between categories of algebras over Lawvere theories.
In this entry, generating functional redirects to generating function. This use does not seem to match the context.
some minimum, for the moment mostly to record this item:
added some formatting and some cross-links (nilpotent groups!) and added pointer to:
stub at locally compact locale
Added:
Introducing hyperstonean spaces:
Added to T-duality a section with the discussion of the usual path-integral heuristics for why the two sigma-models on T-dual backgrounds yield equivalent quantum field theories.
splitting this off from su(2)-anyons: Copied much of the material over, but also added a few more sentences.
For the moment this entry is a cautionary tale about confirmation bias more than an entry about physics.
am in the process of adding some notes on how the D=5 super Yang-Mills theory on the worldvolume of the D4-brane is the double dimensional reduction of the 6d (2,0)-superconformal QFT in the M5-brane.
started a stubby double dimensional reduction in this context and added some first further pointers and references to M5-brane, to D=5 super Yang-Mills theory and maybe elsewhere.
But this still needs more details to be satisfactory, clearly.
Just noticed that we have a duplicate page Jon Sterling.
I have now moved the (little but relevant) content (including redirects) from there to here.
Unfortunately, the page rename mechanism seems to be broken until further notice, therefore I am hesitant to clear the page Jon Sterling completely, for the time being.