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Littlewood-Richardson coefficients are both multiplicities of $GL_n$ tensor products, and of restrictions of $GL_{m+n}$ representations to $GL_m \times GL_n$. I want to turn this equality of numbers into an equality of maps of vector spaces, and generalize it to quantum groups. In more detail:

$\def\Part{\mathrm{Part}} \def\Hom{\mathrm{Hom}}$Let $\Part$ be the abelian category whose objects are collections of finite dimensional vectors spaces $(W_{\lambda})$, one for each partition $\lambda$, such that all but finitely many of the $W_{\lambda}$ are $0$. Here $$\Hom_{\Part}((W_{\lambda}), (X_{\lambda})) = \bigoplus_{\lambda} \Hom_{\mathrm{Vect}}(W_{\lambda}, X_{\lambda})$$ and composition is defined in the obvious manner. I'll write $[\lambda]$ for the element of $\Part$ which is $\mathbb{C}$ at $\lambda$ and $0$ everywhere else.

There are two potential tensor structures on $\Part$, which I will call $\otimes^T$ (for tensor) and $\otimes^R$ (for restriction).

Let $V_{\lambda}(n)$ denote the representation of $GL_n$ with highest weight $\lambda$. In $\otimes^T$, $$\left( [\lambda] \otimes^T [\mu] \right)_{\nu} = \Hom_{GL_n}(V_{\nu}(n), V_{\lambda}(n) \otimes V_{\mu}(n)) \ \mathrm{for} \ n\gg 0.$$ Part of the claim is that, for $n$ sufficiently large, the vector spaces which we say on the right hand side are isomorphic, and this isomorphism can be chosen to commute with the obvious commutator and associator. The definition of $\otimes^T$ for other objects of $\Part$ is forced by linearity.

Similarly, $$\left( [\lambda] \otimes^R [\mu] \right)_{\nu} = \Hom_{GL_m \times GL_n}(V_{\lambda}(m) \boxtimes V_{\mu}(n), V_{\nu}(m+n)) \ \mathrm{for}\ m,n \gg 0.$$

I believe it is not too bad to show that these two tensor structures are equivalent (including the commutator and associator) by identifying them both with the same thing in the representation theory of the symmetric group, through Schur-Weyl duality, although I haven't actually done this.

I think the same statement is true for $GL_n$ quantized at generic $q$. Am I right? Does someone know a reference for this?

Actually, what I really want is the corresponding statement in the category of crystals, so "at $q=0$". What this turns into concretely is that there is a bijection between

  1. $\lambda$-dominant tableaux of shape $\mu$ and content $\nu - \lambda$ and
  2. Yamnouchi tableaux of shape $\nu/\lambda$ and content $\mu$

which commutes with certain jdt moves. I need certain specific cases of this, and I have now checked most of the cases I need by hand -- but I'd much rather cite a general theorem about the category of crystals than write out a lot of tableaux bijections.


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I think the right place to look for the $q=1$ case is in papers by Roger Howe, with branching or tensor product in the title. front.math.ucdavis.edu/… They're based on the $GL(m)\times T^n$ action on ${\mathbb A}^{mn}$, not quite Schur-Weyl duality, but I still expect it should generalize to generic $q$. –  Allen Knutson Jun 25 '12 at 17:45
ncatlab.org/nlab/show/Schur+functor is probably appropriate to mention here. –  David Roberts Jul 3 '12 at 2:32
In particular, there are 5 monoidal structures on the category of Schur functors, which give rise to many interesting, known structures in representation theory. –  David Roberts Jul 3 '12 at 2:34
Thanks! I am trying to digest the five structures now. (1) is $\bigoplus$; it seems boring to mention that as a monoidal structure since it is built into the additive structure. (2) is what I called $\otimes^T$. (5) is Plethysm, which is very interesting, but not the subject of this question. That leaves (3) and (4). I suspect that (3) is $\otimes^R$, but I don't understand Day convolution yet. There should also be some relation between (4) and Plethysm, but I don't have the details. –  David Speyer Jul 3 '12 at 11:08
Sorry, (4) should be related to Kronecker product, not Plethysm. –  David Speyer Jul 3 '12 at 11:41
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1 Answer

A lot, perhaps all, of the crystal case is in Pak and Vallejo's papers PV1 and PV2 (full citations below):

Pak and Vallejo always work with high weight elements of the crystal, also known as Yamanouchi tableaux or tableaux with lattice reading word. I've been finding that everything is much prettier if we also work with the low weight elements. In terms of reading words, these are the ones where the reversal of the reading word has the anti-lattice property.

Let $LR_{+}(\alpha/\beta, \gamma)$ denote high-weight elements of shape $\alpha/\beta$ and content $\gamma$, and let $LR_{-}$ denote the low-weight ones. Let $\alpha \circ \beta$ denote the concatenation of $\alpha$ and $\beta$ (see Figure 1 in PV2). Note that, if $T$ is high weight of shape $\alpha \circ \beta$, for straight shapes $\alpha$ and $\beta$ then the $\alpha$ part is canonical (as PV2 uses the term), so we identify $LR_{+}(\lambda \circ \mu, \nu)$ with certain tableaux of shape $\mu$. These are the so-called $\lambda$-dominant tableaux of shape $\mu$ and content $\nu - \mu$. Similarly, $LR_{-}(\lambda \circ \mu, \nu)$ can be identified with certain tableaux of shape $\mu$.

Let $CF_{+}(\lambda, \mu, \nu)$ and $CF_{-}(\lambda, \mu, \nu)$ denote the tableaux which are in bijection with $LR_{+}(\lambda \circ \mu, \nu)$ and $LR_{-}(\lambda \circ \mu, \nu)$ as in the above paragraph. The latter is probably related somehow to $CF^{\ast}$ in PV2, but I haven't figured out how.

Let $HIVE(\lambda, \mu, \nu)$ denote hives whose boundary in cyclic order is $\lambda$, $\mu$, $-\nu^{ast}$. (Here $(\nu_1, \nu_2, \ldots, \nu_n)^{\ast} = (-\nu_n, \ldots, -\nu_2, - \nu_1)$.)

There are easy bijections (PV1, or simple extrapolation from the maps in PV1) between the following:

  1. $HIVES(\lambda, \mu, \nu)$
  2. $LR_{+}(\nu/\lambda, \mu)$
  3. $LR_{-}(\nu/\mu, \lambda)$
  4. $CF_{-}(\lambda, \mu, \nu)$
  5. $CF_{+}(\lambda, \mu, \nu)$

One can switch $\lambda$ and $\mu$ to get five more sets in easy bijection.

The commutor: The main result of HK is to construct a bijection between $HIVES(\lambda, \mu, \nu)$ and $HIVES(\mu, \lambda, \nu)$ and verify that it is consistent with an $\otimes^T$ commutor they call the cactus commutor.

In fact, I think the cactus commutor also appears in PV2 as the map $\rho_2$, although they don't label it as that. Let $\omega$ be the map on a crystal that applies $w_0$ to the content and reverses the crystal operators. The definition of the cactus commutor is: $u \otimes v \mapsto \omega (\omega(v) \otimes \omega(u))$. Now, if $u \otimes v \in LR_{+}(\lambda \circ \mu, \nu)$, then $\omega(v) \otimes \omega(u)$ will be in $LR_{-}(\mu \circ \lambda, \nu)$. When, as in this case, $\lambda$ and $\mu$ have straight shapes, $\omega$ is just the Shutzenberger operator. Now, remember that only the tableaux of shape $\mu$ is of interest in $LR_{+}(\lambda \circ \mu, \nu)$ and $LR_{-}(\mu \circ \lambda, \nu)$, so what we are looking at is the Schutzenberger operator from $CF_{+}(\lambda, \mu, \nu)$ to $CF_{-}(\mu, \lambda, \nu)$. I suspect that the outer application of $\omega$ is what takes us from $CF_{-}(\mu, \lambda, \nu)$ to $CF_{+}(\mu, \lambda, \nu)$

The composition $$LR_{+}(\nu/\lambda, \mu) \to CF_{+}(\mu, \lambda, \nu) \overset{\omega}{\longrightarrow} CF_{-}(\lambda, \mu, \nu) \to LR_{+}(\nu/\mu, \lambda)$$ is what PV2 call $\rho_2$.

In the meanwhile, PV2 also have a commutor for $\otimes^R$: The map they call $\rho_1$, which goes from $LR_{+}(\nu/\lambda, \mu)$ to $LR_{+}(\nu/\mu, \lambda)$ is what I would consider the most natural possible commutor for $\otimes^R$.

According to footnote 6 in PV2, the equality of $\rho_1$ and $\rho_2$ is the main result of DK.

The associator PV2 also discusses associators, under the name "octahedron maps". HK show that the HIVE-based associator, introduced in KTW, is isomorphic to the $\otimes^T$ associator.

Some experimentation suggests that the compatible thing on the $\otimes^R$ side is a map $$\bigcup_{\rho} LR_{+}(\kappa, \lambda, \rho) \times LR_{-}(\rho, \mu, \nu) \longleftrightarrow \bigcup_{\sigma} LR_{+}(\kappa, \sigma, \nu) \times LR_{-}(\lambda, \mu, \sigma)$$ and that this map should be the map PV2 call $\zeta$ -- the "tableaux switching" map. Section 7.3 of PV2 produces a similar map using high weight tableaux everywhere, with four uses of $\zeta$. I think that three of those uses are just to switch between $LR_{+}$ and $LR_{-}$ as needed.

Conjecture 3 of PV2 is that their associator and the HIVE based associator match. It is not clear to me whether or not their footnote 8 is claiming that HK prove conjecture 3 but, if they do, I don't see it.

PV1 Pak and Vallejo, Combinatorics and geometry of Littlewood-Richardson cones, European J. Combin. 26 (2005), no. 6, 995–1008

PV2 Pak and Vallejo, Reductions of Young tableau bijections, SIAM J. Discrete Math. 24 (2010), no. 1, 113–145

HK Henriques and Kamnitzer, The octahedron recurrence and $\mathfrak{gl}_n$ crystals Adv. Math. 206 (2006), no. 1, 211–249

DK Danilov and Koshevoy Massifs and the combinatorics of Young tableaux, Uspekhi Mat. Nauk 60 (2005), no. 2(362), 79--142; translation in Russian Math. Surveys 60 (2005), no. 2, 269–334

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