Heisenberg algebra, wedges and crystals

Thomas Gerber

doi:10.1007/s10801-018-0820-8

Journal of Algebraic Combinatorics

February 2019, Volume 49, Issue 1, pp 99–124 | Cite as

Heisenberg algebra, wedges and crystals

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Thomas Gerber

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First Online: 19 March 2018

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Abstract

We explain how the action of the Heisenberg algebra on the space of q-deformed wedges yields the Heisenberg crystal structure on charged multipartitions, by using the Boson–Fermion correspondence and looking at the action of the Schur functions at $q=0$. In addition, we give the explicit formula for computing this crystal in full generality.

Keywords

Fock space Categorification Quantum groups Heisenberg algebra Crystals Symmetric functions Combinatorics

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Notes

Acknowledgements

Many thanks to Emily Norton for pointing out an inconsistency in the first version of this paper and for helpful conversations.

Appendix A: Crystal level-rank duality

We recall, using a slightly different presentation, the results of [9, Section 4] concerning the crystal version of level-rank duality.

There is a double affine quantum group action on the wedge space. In Sect. 2.2.1, we have explained how $\mathcal {U}'_q (\widehat{\mathfrak {sl}_e})$ acts on $\mathcal {F}_s$. It turns out that $\mathcal {U}'_p (\widehat{\mathfrak {sl}_\ell })$, where $p=-1/q$, acts on $\mathcal {F}_s$ in a similar way. More precisely we will:

(1)

index ordered wedges by charged e-partitions, using an alternative bijection $\dot{\tau }$,
(2)

make $\mathcal {U}'_p (\widehat{\mathfrak {sl}_\ell })$ act on $\mathcal {F}_s$ via this new indexation by swapping the roles of e and $\ell $ and replacing q by p.

To define $\dot{\tau }$, recall that we have introduced the quantities $z(n)\in \mathbb {Z}$, $1\le y(n) \le \ell $ and $1\le x(n)\le e$ for each $n\in \mathbb {Z}$. To each pair $(1,c)\in \{1\}\times \mathbb {Z}$, we associate the pair $\dot{\tau }(1,d)=(j,d)\in \{1,\dots ,\ell \}\times \mathbb {Z}$ where

$$\begin{aligned} j=x(-c)\quad \text {and}\quad d=-(y(-c)-1)+\ell z(-c). \end{aligned}$$

In particular, $\dot{\tau }$ sends the bead in position (1, c) into the rectangle $z(-c)$, on the row $x(-c)$ and column $y(-c)$ (numbered from right to left within each rectangle).

The map $\dot{\tau }$ is bijective and we can see $\dot{\tau }^{-1}$ as the following procedure:

(1)

Divide the $\ell $-abacus into rectangles of size $e\times \ell $, where the z-th rectangle ($z\in \mathbb {Z}$) contains the positions (j, d) for all $1\le j\le \ell $ and $-e+1+ze\le d \le ze$.
(2)

Relabel each (j, d) by the second coordinate of $\dot{\tau }^{-1}(j,d)$, see Fig. 4 for an example.
(3)

Replace the newly indexed beads on a 1-abacus according to this new labeling.

Fig. 4
Relabelling bead positions in the e-abacus according to $\dot{\tau }^{-1}$, for $\ell =4$ and $e=3$. One should compare this with the labeling of Fig. 1

We see that $\dot{\tau }^{-1}$, so also $\dot{\tau }$, actually only depends on e, not on $\ell $. In fact, explicit formulas for $\dot{\tau }$ and $\dot{\tau }^{(1)}$ are given by

$$\begin{aligned} \dot{\tau }(1,c)= & {} \left( (-c\mod e) + 1 \, , \, -\left\lfloor \frac{-c}{e}\right\rfloor \right) \\= & {} \left( (-c\mod e) + 1 \, , \, \frac{-c-(-c\mod e)}{e} \right) , \end{aligned}$$

and

$$\begin{aligned} \dot{\tau }^{-1} (j,d) = \left( 1 \, , \, -(j-1)+ed \right) , \end{aligned}$$

and one notices that $\dot{\tau }$ corresponds to taking the usual e-quotient and the e-core of a partition, where the e-core is encoded in the e-charge. More precisely, the renumbering of the beads according to $\dot{\tau }$ is the well-know “folding” procedure used to compute the e-quotient, see [13].

Definition A.1

The (twisted) level-rank duality is the bijective map $\dot{\tau }\circ (.)'\circ \tau ^{-1}$

Remark A.2

The map $\dot{\tau }\circ \tau ^{-1}$ already defines a level-rank duality, this was the one studied by Uglov [27]. However, the twisted version defined above (i.e. where the conjugation is added) is the one that is relevant when it comes to crystals, see Theorem A.4 below.

There is a convenient way to picture the crystal level-rank duality as follows. Starting from an $\ell $-abacus, stack copies on top of each other by translating by e to the right. Then, extract a vertical slice of the resulting picture, and read off the corresponding e-partition by looking at the white beads (instead of black beads) in each column, starting from the leftmost one.

Example A.3

The abacus $\mathcal {A}(\varvec{\lambda },\mathbf {s})$ with $\ell =4$, $e=3$, $\varvec{\lambda }=(1,\emptyset ,1^3,5)$ and $\mathbf {s}=(-1,-1,1,1)$ looks as follows (the origin is represented with the boldfaced vertical bar). Open image in new window

Stacking copies of $\mathcal {A}(\varvec{\lambda },\mathbf {s})$ gives Open image in new window

and extracting one vertical e-abacus yields Open image in new window

which corresponds to $|(1,2.1^2,2.1^4),(-1,1,0)\rangle $.

We have the following induced maps

Open image in new window

Like $\tau $, the bijection $\dot{\tau }$ induces a $\mathcal {U}'_p (\widehat{\mathfrak {sl}_\ell })$-module isomorphism, and $\mathcal {F}_s$ has a $\mathcal {U}'_p (\widehat{\mathfrak {sl}_\ell })$-crystal, given by the same rule as the $\mathcal {U}'_q (\widehat{\mathfrak {sl}_e})$-crystal by swapping the roles of e and $\ell $ and replacing q by p.

Theorem A.4

Via level-rank duality,

(1)

the $\mathcal {U}'_q (\widehat{\mathfrak {sl}_e})$-action and the $\mathcal {U}'_p (\widehat{\mathfrak {sl}_\ell })$-action on $\mathcal {F}_s$ commute, and
(2)

the $\mathcal {U}'_q (\widehat{\mathfrak {sl}_e})$-crystal and the $\mathcal {U}'_p (\widehat{\mathfrak {sl}_\ell })$-crystal of $\mathcal {F}_s$ commute.

Proof

The first point is essentially due to Uglov [27, Proposition 4.6], where he uses the non-twisted version of level-rank duality, see [9, Theorem 3.9] for the justification in the twisted case. The second point is [9, Theorem 4.8]. $\square $

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Thomas Gerber
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1.Lehrstuhl D für MathematikRWTH Aachen UniversityAachenGermany

Heisenberg algebra, wedges and crystals

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