Equations defining the affine Grassmannian of $\SL_n$

A solved problem: Standard monomials on the finite Grassmannian

Let $V = \bbk^n$. The Plücker embedding realises the finite Grassmannian as a projective variety: $$ \Gr(r, n) = \{ W \subseteq V \mid \dim W = r\} \xinjto{p} \bbP(\wedge^r V), \quad \Span_\bbk\{v_1, \ldots, v_r\} \mapsto [v_1 \wedge \cdots \wedge v_r]. $$ Coordinates on $\bbP(\wedge^r V)$ are labelled by the set $C_{r, n} = \{ I \subseteq \{1, \ldots, n\} \mid |I| = r\}$ of $r$-element subsets: $$ \bbk[\bbP(\wedge^r V)] = \bbk[x_I \mid I \in C_{r, n}] , \quad \text{ where } I = \{i_1 < \cdots < i_r\} \text{ and } x_I \text{ is dual to } e_{i_1} \wedge \cdots \wedge e_{i_r}. $$

The purpose of standard monomial theory is to describe a $\bbk$-basis of the homogeneous coordinate ring $\bbk[\Gr(r, n)] = \bbK[x_I \mid I \in C_{r, n}] / \cP$, where $\cP = \ker p^*$ is the Plücker ideal.

The monomial $x_I x_J x_K \in \bbk[x_I \mid I \in C_{r, n}]$ is a standard monomial if $I \leq J \leq K$ entrywise, (as a tableau, this means weakly increasing down the columns). Of course there are non-standard monomials, say if $I = \{1, 3, 6, 7\}$ and $J = \{2, 3, 4, 8\}$:

$I$ and $J$ are incomparable under $\leq$ (the problem is highlighted pink in the diagram) and so cannot be part of a standard monomial $x_I x_J x_K$. We will straighten $x_I x_J$ by finding a quadratic relation $P_{I, J} \in \cP$ that contains $x_I x_J$ and vanishes on the embedded Grassmannian $\Gr(r = 4, n)$.

Split $(I, J)$ into $A = (1, 3)$, $B = (2, 3, 4, 6, 7)$ and $C = (8)$ as above, and send $x_A \otimes x_B \otimes x_C$ through the map $$ \wedge^2 V \otimes \wedge^5 V \otimes \wedge^1 V \xinjto{1 \otimes \mathsf{comult}_{2, 3} \otimes 1} \wedge^2 V \otimes \wedge^2 V \otimes \wedge^3 V \otimes \wedge^1 V \xto{\mathsf{mult}_{2, 2} \otimes \mathsf{mult}_{3, 1}} \wedge^4 V \otimes \wedge^4 V \surjto \Sym^2(\wedge^4 V) $$ to get a quadratic relation $P_{I, J}$ which includes $x_I x_J$. ($\mathsf{comult}$ is the signed unshuffling of the sequence): $$ \begin{aligned} x_{13} \otimes x_{23467} \otimes x_{8} &\mapsto x_{13} \otimes ( x_{23} \otimes x_{467} - x_{24} \otimes x_{367} + x_{26} \otimes x_{347} - \cdots + x_{67} \otimes x_{234} ) \otimes x_{8} \\ &\mapsto 0 + x_{1234} x_{3678} - x_{1236} x_{3478} - \cdots + \underbrace{x_{1367} x_{2348}}_{x_I x_J} = P_{I, J} \end{aligned} $$ $P_{I, J}$ vanishes on $\Gr(r, n)$ because of the $\wedge^{r+1}$ term coming from $x_A$, hence $P_{I, J} \in \cP$. A more detailed inductive argument shows that any monomial $x_{I_1} x_{I_2} \cdots x_{I_\ell}$ can be straightened to a linear combination of standard monomials, hence the standard monomials span the ring $\bbk[x_I \mid I \in C_{r, n}] / \cP$. A more careful argument shows they are linearly independent.

Our problem: Standard monomials on the affine Grassmannian $\Gr_{\SL_n}$

The affine Grassmannaian $\Gr_{\SL_n}$ admits an embedding $i_n$ into the infinite Grassmannian $\Gr(\infty)$, which in turn embeds via the Plücker embedding $p$ into the projectivisation $\bbP(\cF)$ of Fock space. Drawing analogies from above, $\Gr(\infty)$ is like $\Gr(r, n)$ and $\cF$ is like $\wedge^r V$, however $\Gr_{\SL_n}$ is quite a different object. $$ \Gr_{\SL_n} \xinjto{i_n} \Gr(\infty) \xinjto{p} \bbP(\cF) \quad \quad $$ The ideal $\cP$ cutting out $\Gr(\infty)$ inside $\bbP(\cF)$ is an infinite analogue of the Plücker relations. By a conjecture of Kreiman, Lakshmibai, Magyar, and Weyman [KLMW07] recently proven by Muthiah, Weekes, and Yacobi [MWY18], the set $\cS_n$ of linear functions on $\cF$ vanishing on $\Gr_{\SL_n}$ are given by the shuffle equations.

Maya diagrams, semi-infinite wedges, and charged partitions

A Maya diagram $\mathsf{m} \colon \bbZ \to \{\circ, \bullet\}$ is a 2-colouring that is eventually white to the left and black to the right.

It can be recorded by the location of its white beads $\mathsf{m}^\circ \colon \bbZ_{<0} \to \bbZ$, or its black beads $\mathsf{m}^\bullet \colon \bbZ_{\geq 0} \to \bbZ$.

$$ \mathsf{m}^\circ = (\ldots, -6, -5, -4, -2, -1, 2, 4) \, \mid \, (-3, 0, 1, 3, 5, 6, 7, \ldots) = \mathsf{m}^\bullet$$

The union $\mathsf{m}^\circledcirc \colon \bbZ \to \bbZ$ is a bijection, where $\mathsf{m}^\circledcirc(i) - i$ stabilises to the charge ${\color{blue} c(\mathsf{m})}$ (here ${\color{blue} c(\mathsf{m})} = 1$).

The sequence $-(\mathsf{m}^\circledcirc(i) - i - {\color{blue} c(\mathsf{m})})$ defines a partition $({\color{purple} 4}, {\color{purple} 2}, {\color{purple} 2}, {\color{purple} 1}, 0, 0, 0, \ldots)$. The following are in bijection:

1. The Maya diagram $\mathsf{m} \colon \bbZ \to \{ \circ, \bullet \}$ shown above, 2-colouring the integers.
2. The semi-infinite wedge $e_{-3} \wedge e_0 \wedge e_1 \wedge e_3 \wedge e_5 \wedge e_6 \wedge \cdots$ giving the sequence $\mathsf{m}^\bullet$.
3. The charged partition $({\color{blue} c}, {\color{purple} \lambda}) = ({\color{blue} 1}, {\color{purple} (4, 2, 2, 1)})$.

These three combinatorial objects all label the same basis of Fock space $\cF$.

Fermionic Fock space

The Fermionic Fock space $\cF$ is the vector space with basis given by Maya diagrams (or semi-infinite wedges, or charged partitions). It is graded by charge: $$ \cF = \bigoplus_{c \in \bbZ} \cF^{(c)}, \quad \text{ where } \cF^{(c)} = \Span_\bbk\{ (c, \lambda) \mid \lambda \in \mathsf{Partitions} \}. $$ The homogeneous coordinate ring is a polynomial ring in infinite variables: $\bbk[\bbP(\cF)] = \bbk[x_\mathsf{m} \mid \mathsf{m} \in \mathsf{Mayas}]$. Similarly to the finite case, we say that $x_{\mathsf{m}_1} \cdots x_{\mathsf{m}_\ell}$ is a standard monomial if $\mathsf{m}_1 \leq \cdots \leq \mathsf{m}_\ell$, where the ordering $\leq$ is by containment of charged partitions.

The standard monomials form a $\bbk$-basis of $\bbk[\Gr(\infty)]$, however they do not appear to play nicely when the shuffle relations $\cS_n$ are also introduced.

The infinite (or Sato) Grassmannian

Define a vector space $F^\infty = \Span_\bbk\{e_i \mid i \in \bbZ\}$ and distinguished subspaces $F^{\geq i} = \Span_\bbk\{e_j \mid j \geq i\}$.

A subspace $V \subseteq F^\infty$ is virtual if $F^{\geq n} \subseteq V \subseteq F^{\geq -n}$ for some $n \geq 0$.

The infinite Grassmannian $\Gr(\infty) = \{ V \subseteq F^\infty \mid V \text{ is virtual} \}$ is the set of virtual subspaces of $F^{\infty}$.

The Plücker embedding $\Gr(\infty) \xinjto{p} \bbP(\cF)$ forms the semi-infinite wedge of a virtual space. $$ \left(F^{\geq n} \subseteq V \subseteq F^{\geq -n}\right) \mapsto \bigwedge^{\mathrm{top}} \left(V/F^{\geq n}\right) \wedge e_n \wedge e_{n+1} \wedge \cdots $$ The relative charge of $V, W \in \Gr(\infty)$ with $F^{\geq n} \subseteq V, W \subseteq F^{\geq -n}$ is $$ \relcharge(V, W) = \dim(V / F^{\geq n}) - \dim(W / F^{\geq n}), $$ while the charge of $V$ is its relative charge to $F^{\geq 0}$, written $ c(V) = \relcharge(V, F^{\geq 0}). $ The Plücker embedding respects charge: $\Gr(\infty)^{(c)} \xinjto{p} \bbP(\cF^{(c)})$.

The Plücker ideal

For $d \geq 0$ define a quadratic map (and check the sum is well-defined!) $$ \Omega_d \colon \cF^{(c)} \to \cF^{(c + d)} \otimes \cF^{(c - d)} ,\quad \quad \Omega_d(\omega) = \sum_{I \subseteq \bbZ, |I| = d} \psi_I(\omega) \otimes \psi_I^*(\omega), $$ where $\psi_I$ means $\psi_{i_1} \circ \cdots \circ \psi_{i_d}$ for $I = \{i_1 < \cdots < i_d\}$. The Plücker ideal $\cP$ is the set of equations formed by postcomposing the $\{\Omega_d \mid d \geq 0\}$ with coordinate functions on $\cF^{(c + d)} \otimes \cF^{(c - d)}$, for $d \geq 0$.

The action of $\widehat{\fsl_n}$ on Fock space, the representation $V(\Lambda_0)$

The action of $\widehat{\fsl_n}$ on the charged partition $(c, \lambda)$ examines its residues:

The Chevalley generators $E_{\color{orange} \bullet}, E_{\color{cyan} \bullet}, E_{\color{purple} \bullet}$ remove boxes of the their colour, without modifying the charge:

The Chevalley generators $F_{\color{orange} \bullet}, F_{\color{cyan} \bullet}, F_{\color{purple} \bullet}$ add boxes their colour, without modifying the charge:

The derivation $d$ acts on $(c, \lambda)$ by counting boxes of its colour (purple), so $d$ scales our example by 2.

In terms of the Clifford operators, we have $E_i = \sum_{j \in i + n \bbZ} \psi_{j - 1} \psi_j^*$ and $F_i = \sum_{j \in i + n \bbZ} \psi_j \psi_{j - 1}^*$.

The basic representation $V(\Lambda_0)$ of $\widehat{\fsl}_n$ is the submodule of $\cF$ generated by the charge zero empty partition: $$ V(\Lambda_0) = U(\widehat{\fsl}_n) \cdot (0, \varnothing) \subseteq \cF^{(0)}. $$ The shuffle relations $\cS_n$ cut out $V(\Lambda_0)$ inside $\cF$.

Clifford operators on Fock space

The Clifford operators $\psi_i, \psi_i^* \colon \cF \to \cF$ form the wedge or interior product with $e_i$. $$ \psi_i(\omega) = e_i \wedge \omega, \quad \psi_i^*(\omega) = \iota_{e_i}(\omega) $$ In terms of Maya diagrams, $\psi_i \mathsf{m}$ turns the $i$th bead of $\mathsf{m}$ black ($\psi_i \mathsf{m} = 0$ if it is already black) and multiply by a sign depending on the number of black beads to the left of $i$. With the $\mathsf{m}$ shown above, $\psi_1 \mathsf{m} = 0$ while $\psi_2 \mathsf{m}$ is the negative of the following diagram:

$\psi_i^*$ acts similarly after swapping white with black. The Clifford operators are graded: $$ \cdots \xtofrom[\psi_i^*]{\psi_i} \cF^{(-1)} \xtofrom[\psi_i^*]{\psi_i} \cF^{(0)} \xtofrom[\psi_i^*]{\psi_i} \cF^{(1)} \xtofrom[\psi_i^*]{\psi_i} \cdots $$

The shuffle equations

For $I \subseteq \bbZ$ and $n \in \bbZ$, set $I + n = \{i + n \mid i \in I\}$. For $d \geq 1$, define the linear map $$ \sh_d^{n} \colon \cF \to \cF, \quad \quad \sh_d^{n} = \sum_{I \subseteq \bbZ, |I| = d} \psi_{I + n} \circ \psi_{I}^* $$

The shuffle ideal $\cS_n \subseteq \bbk[\bbP(\cF^{(0)}]$ cutting out the $\widehat{\fsl}_n$ representation $V(\Lambda_0) \subseteq \cF^{(0)}$ is $\cS_n = \sum_{d \geq 1} \im \sh_d^n$.

FLOTW multipartitions and standard monomials

By a theorem of Kostant, $\bbk[\Gr_{\SL_n}] \cong \bigoplus_{r \geq 0} V(r \Lambda_0)^*$, with the Cartan product as the algebra structure on the right. The work of [FLOTW99] describes a basis for $V(r \Lambda_0)$ in terms of FLOTW multipartitions, an $r$-tuple of partitions satisfying containment and $n$-cylindricity:

Above is an $(r = 4)$-multipartition $\pmb{\lambda}$ satisfying containment and $(n=3)$-cylindricity. To be FLOTW, the union of residues $\Res(\ell, \pmb{\lambda})$ for each length $\ell$ row needs to be incomplete, for all $\ell > 1$. For $\pmb{\lambda}$ above:

and hence $\pmb{\lambda}$ is not a FLOTW multipartition, as both $\Res(2, \pmb{\lambda})$ and $\Res(1, \pmb{\lambda})$ are complete.

Our plan

In the finite Grassmnannian $\Gr(k, n)$ of $k$-planes in $n$-space, extracting certain relations from the Plücker ideal $\cP$ lead to a straightening rule in the coordinate ring $\bbk[\bbP(\bigwedge^k \bbk^n)]$, rewriting arbitrary monomials in terms of standard monomials: multipartitions satisfying the containment relation above.

In the infinite case we have both the Plücker ideal $\cP$ and the shuffle ideal $\cS_n$, and we are aiming to find a straightening law to rewrite monomials (multipartitions) into FLOTW multipartitions.

References

[KLMW07]

V. Kreiman, V. Lakshmibai, P. Magyar, and J. Weyman, "On ideal generators for affine Schubert varieties", Algebraic groups and homogeneous spaces, Tata Inst. Fund. Res. Stud. Math. 19 (2007), 353-388.

[MWY18]

D. Muthiah, A. Weekes, and O. Yacobi, "The equations defining affine Grassmannians in type $A$, arXiv:1708.07076v2.

[FLOTW99]

O. Foda, B. Leclerc, M. Okado, J. Thibon, and T Welsh, "Branching functions of $A^{(1)}_{n−1}$ and Jantzen-Seitz problem for Ariki-Koike algebras", .Adv. Math., 141:322–365, 1999.