Equations defining the affine Grassmannian of $\SL_n$

Joel Gibson

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.

Problem: Confirm that $\cS_n$ is the defining ideal of $\Gr_{\SL_n}$ inside $\Gr(\infty)$.

Approach: Develop a standard monomial theory for $\bbk[\Gr(\infty)] / \cS_n$, and compare with a known basis for $\bbk[\Gr_{\SL_n}]$ given by FLOTW multpartitions.

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 ${\color{blue} c(\mathsf{m})} = 1$).

$i$ $\cdots$ $-6$ $-5$ $-4$ $-3$ $-2$ $-1$ $0$ $1$ $2$ $3$ $4$ $5$ $6$ $\cdots$
$\mathsf{m}^\circledcirc(i)$ $\cdots$ $-5$ $-4$ $-2$ $-1$ $2$ $4$ $-3$ $0$ $1$ $3$ $5$ $6$ $7$ $\cdots$
$\mathsf{m}^\circledcirc(i) - i$ $\cdots$ $1$ $1$ $2$ $2$ $4$ $5$ $-3$ $-1$ $-1$ $0$ $1$ $1$ $1$ $\cdots$
$\mathsf{m}^\circledcirc(i) - i - {\color{blue} c(\mathsf{m})}$ $\cdots$ $0$ $0$ $1$ $1$ $3$ $4$ $-4$ $-2$ $-2$ $-1$ $0$ $0$ $0$ $\cdots$

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 Lie algebra $\widehat{\fsl_n}$ is the Kac-Moody algebra associated to a cycle diagram on $n$ nodes. For example, $\widehat{\fsl_3}$ is generated by the Chevalley generators $E_{\color{orange} \bullet}$, $E_{\color{cyan} \bullet}$, $E_{\color{purple} \bullet}$, $F_{\color{orange} \bullet}$, $F_{\color{cyan} \bullet}$, $F_{\color{purple} \bullet}$, and the derivation ${\color{purple} d} \in \fh$ satisfying $[{\color{purple} d}, E_i] = \delta_{i, {\color{purple} \bullet}} E_i$.

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

Take a charged partition $(c, \lambda) = (1, (4, 2, 2, 1))$ Assign each cell its content, shifted by the charge $c$ Reduce modulo $n$ to find the 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:

$$ \xto{E_{\color{orange} \bullet}}$$
$$ + $$

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

$$ \xto{F_{\color{purple} \bullet}}$$
$$ + $$
$$ + $$

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:

$$ \supseteq $$
$$ \supseteq $$
$$ \supseteq $$
$$ \supseteq $$

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:

$\ell$ $6$ $5$ $4$ $3$ $2$ $1$
$\Res(\ell, \pmb{\lambda})$ $\{{\color{cyan} \bullet}\}$ $\{\purple{\bullet}\}$ $\{{\color{cyan} \bullet}, \orange{\bullet}\}$ $\{{\color{cyan} \bullet}\}$ $\{{\color{cyan} \bullet}, \orange{\bullet}, \purple{\bullet}\}$ $\{{\color{cyan} \bullet}, \orange{\bullet}, \purple{\bullet}\}$

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.


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