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just a stub, for the moment just so as to satisfy a link that had long been requested at Schur function
added pointer to these textbook accounts:
William Fulton, Joe Harris, Section 6.1 of: Representation Theory: a First Course, Springer, Berlin, 1991 (doi:10.1007/978-1-4612-0979-9)
Ambar N. Sengupta, Section 10.2 of: Representing Finite Groups – A Semisimple Introduction, Springer 2012 (doi:10.1007/978-1-4614-1231-1)
added this pointer:
Interesting closing paragraph
Schur-Weyl duality has been a surprisingly effective technical tool in gauge-string duality, capturing crucial aspects of the map between gauge theory states and spacetime string theory states, both for two dimensional and four dimensional gauge theory. It is undoubtedly going to continue to play this role and provide valuable information on many interesting physical questions on gauge theory, especially in relation to its stringy spacetime dual. It is natural to wonder if an appropriately enriched version of Schur-Weyl duality might actually give a complete mathematical expression of the background independent content of gauge string duality.
Yes, there is something to be unearthed for us here.
By the way, it only just now occurs to me what it is that the hook-content formula measures in analogy to how the hook length formula gives the dimension of the irreps of $Sym(n)$: The hook-content formula gives the dimension of the irreps of $SU(n)$!
I am still looking for a canonical reference for this, but one place where it is at least stated clearly is p. 2 in these notes.
Coincidently I happen to have on my desk the book she refers to
[5] S. Sternberg, Group Theory and Physics, Cambridge University Press, Cambridge, 1994.
C. 27 on p. 352 gives the dimension of a rep of $GL(V)$ as the RHS of the hook-content formula.
A book I always meant to read.
Ah, is hook-content hence for both $GL(n)$ and $SU(n)$?
Do we see any of that idea of $q$-deformation at $q = 1$ is $\mathbb{F}_1$-mathematics? After all,
$GL(n,\mathbb{F}_1) \;\simeq\; \Sigma_n.$While we are cross-posting, let me highlight another observation:
This means in particular that the eigenvalues of the Cayley distance kernel at log-integral inverse temperature are (with this formula) of the form
$EigVals[e^{-ln(N) \cdot d_C}]_\lambda \;=\; \frac{n!}{N^n} \frac { dim(U^{(\lambda)}) } { dim(S^{(\lambda)}) }$where $U^{(\lambda)}$ and $S^{(\lambda)}$ are the irreps of $SU(n)$ and $Sym(n)$, respectively.
Better yet, this means that the dependency on the hook-length drops out, and also the factor $n!$ cancels out, and we are left with the “content”
$EigVals[e^{-ln(N) \cdot d_C}]_\lambda \;=\; \frac{1}{N^n} \underset{(i,j)}{\prod} \big( N + j - i \big)$But that is finally the kind of formula you had been conjecturing all along, in the other thread!
That book by Sternberg is fun, but it’s not quite the authorative canonical reference for these matters that I would hope for.
What’s a good maths textbook that proves hook length/hook-content formula as measuring dimensions of irreps?
Something on the relationship between reps of $GL(n)$ and $SU(n)$ by Stanley here. But I must dash.
Looks like some interesting discoveries above.
Incidentally, here is L. Motl, about 6 years ago, speculating that M-theory is somehow given by a sum over Young diagrams.
In any case, we have that the expectation value of any observable in our fundamental-weight-system quantum states is a sum over Young diagrams (namely of the expectation value of that observable in the pure states contained in the mixed state that is the fundamental-weight-system). And as in those matrix model considerations, it is natural for the fundamental-weight-system states to be considered in the limit of number of strands going to infinity, hence in the colimit limit over direct system
$\mathcal{A}^{pb}_2 \hookrightarrow \mathcal{A}^{pb}_3 \hookrightarrow \mathcal{A}^{pb}_4 \hookrightarrow \cdots \,.$Coming back to the question in #8, regarding which reps exactly it is that the hook-content formula counts the dimension of.
From appendix C.7 Sternberg 94 it must be those reps of $GL(n,\mathbb{C})$ which are labeled by Young diagrams.
By Thm. 2 on p. 114 of Fulton 97 these are precisely the polynomial irreps of $GL(n,\mathbb{C})$.
By Section 5.8 in Sternberg 94 these are precisely the irreps of $SL(n,\mathbb{C})$.
By a reference I lost these are the irreps of $SU(n)$ if one discards the Young diagrams with more than $n$ rows, or something like this (?)
Does that sound right?
The first three points tally with what I’ve read.
And the fourth refers to this document:
It is less well-known that the irreducible representations of $SU(n)$ are naturally indexed by partitions with less than $n$ parts.
So I guess discard Young diagrams with more than $(n-1)$ rows.
Thanks, that was the reference.
It’s puzzling how these basic facts are so scattered through the literature, even scattered within dedicated accounts.
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