Abstract
We describe a relationship between work of Gatto, Laksov, and their collaborators on realizations of (generalized) Schubert calculus of Grassmannians, and the geometric Satake correspondence of Lusztig, Ginzburg, and Mirković and Vilonen. Along the way we obtain new proofs of equivariant Giambelli formulas for the ordinary and orthogonal Grassmannians, as well as a simple derivation of the “rim-hook” rule for computing in the equivariant quantum cohomology of the Grassmannian.
DA was partially supported by a postdoctoral fellowship from the Instituto Nacional de Matemática Pura e Aplicada (IMPA) and by NSF Grant DMS-1502201.
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Notes
- 1.
In general, a cocharacter \(\varpi :\mathbb {C}^* \rightarrow T \subseteq G\) determines a reductive subgroup \(G_\varpi \subseteq G\), the centralizer of its image; the corresponding parabolic \(P_\varpi \) is generated by \(G_\varpi \) together with the Borel.
- 2.
When \(n=1\), the spaces are 0-dimensional; when \(n=2\), they coincide with type A spaces. For \(n=3\), there are coincidences \(\mathcal {Q}^4 = {Gr}(2,4)\) and \(OG^+(3,6)=OG^-(3,6)=\mathbb {P}^3\). For \(n=4\), there are also coincidences \(\mathcal {Q}^6 = OG^+(4,8) = OG^-(4,8)\). The reader may use these to verify our claims, but beware that the torus actions are usually written differently.
- 3.
The indexing most natural to our setup is slightly nonstandard. The rows and columns of A(x) are labelled \(n-1,\ldots ,0\) (left to right, top to bottom); similarly, the rows of \(B_I(x|t)\) are labelled \(n-1,\ldots ,0\) (top to bottom) and its columns are labelled \(i_1,\ldots ,i_k\) (left to right).
- 4.
- 5.
Our indexing of t variables is reversed when compared with that of [39], since we use opposite Schubert classes.
- 6.
This “spin basis” is not directly connected to spin representations and the orthogonal Grassmannian, as far as we are aware.
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Acknowledgements
This note is partly based on a talk given by the first author at a conference dedicated to the memory of Dan Laksov (Mittag-Leffler, June 2014). The ideas grew out of conversations we had with Roi Docampo at IMPA. We learned about the connection between quantum Schubert calculus and the Satake correspondence from a remarkable preprint of Golyshev and Manivel [19], and the debt we owe to their work should be evident. We also thank Reimundo Heluani and Joel Kamnitzer for helping us understand the geometric Satake correspondence. Finally, we thank the referee for a very careful reading and thoughtful comments.
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Appendix: Pfaffians and Spinors
Appendix: Pfaffians and Spinors
In this appendix, we prove a change-of-basis formula in the spin representation, where Pfaffians play the role analogous to determinants in the exterior algebra. This refines similar formulas of Chevalley and Manivel [10, 37].
First, for any complex vector space V with symmetric bilinear form \(\langle \;,\;\rangle \), the associated Clifford algebra is the quotient of the tensor algebra by two-sided ideal forcing the relation \(v\cdot w + v\cdot w = \langle v,w\rangle 1\) for all vectors \(v,w\in V\):
(Note that this definition differs slightly from the standard one, where \(\langle v,w \rangle 1\) would be replaced by \(2\langle v,w \rangle 1\)—but it is equivalent, up to rescaling the form \(\langle \;,\; \rangle \), since we are not in characteristic 2.) If the bilinear form is zero, this is just the exterior algebra: \(Cl(V) \cong \textstyle \bigwedge ^\bullet V\). In general, \(\dim Cl(V) = 2^{\dim V}\), with a basis consisting of products \(v_I=v_{i_1}\cdots v_{i_k}\) of distinct basis elements of V. (One can prove this by degenerating to the exterior algebra.)
We note the following general formulas in Cl(V), which are immediate from the defining relations. First, if x and y are orthogonal vectors in V (i.e., \(\langle x,y \rangle = 0\)), then
in Cl(V). Next, suppose \(x = x_1\cdots x_r \in Cl(V)\) is a product of vectors in V such that \(X={{\,\mathrm{span}\,}}\{x_1,\ldots ,x_r\} \subseteq V\) is isotropic. Then for any vector \(y\in X\),
in Cl(V).
From now on, we assume \(\dim V=2n\) and its bilinear form is nondegenerate. Let \(y_{\overline{n-1}},\ldots ,y_{\overline{0}},y_0,\ldots ,y_{{n-1}}\) be an “orthogonal basis”, meaning \(\langle y_{\overline{\imath }}, y_j \rangle = \delta _{ij}\). (We interpret the bar as a negative sign, so \(\overline{\overline{\imath }}=i\).) Let \(y=y_0\cdots y_{n-1}\). The spin representation is the left ideal
A (pure) spinor is an element of the form \(z\cdot y \in \mathbb {S}\), where \(z = z_1\cdots z_n\) is a product of vectors \(z_1,\ldots ,z_n \in V\) which span a maximal isotropic subspace of V.
Let \([m] = \{0,\ldots ,m-1\}\) for any integer \(m\ge 1\). For a subset \(I=\{i_1<\cdots <i_r\} \subseteq [n]\), let \(I' = \{i'_1<\cdots <i'_{n-r}\} = [n]\smallsetminus I\). Given such an I, the corresponding standard spinor is
These form a basis for \(\mathbb {S}\), as I ranges over all subsets of [n]. Note that \(y_\emptyset = y_{\overline{0}}\cdots y_{\overline{n-1}}\cdot y\), and
which is sometimes useful.
The spin representation becomes an \(\mathfrak {so}_{2n}\)-module via the embedding \(\mathfrak {so}_{2n} \cong \textstyle \bigwedge ^2 V \hookrightarrow Cl(V)\) by
So \(\mathfrak {so}_{2n}\) preserves the parity of basis vectors of Cl(V), and the spin representation breaks into two irreducible half-spin representations
Our convention will be that \(\mathbb {S}^+\) is spanned by standard spinors \(y_I\), with I an even subset of [n], i.e., it has even cardinality.
Now let \(x_{\overline{n-1}},\ldots ,x_{\overline{0}},x_0,\ldots ,x_{{n-1}}\) be another orthogonal basis, and assume it is related to the \(y_i\)’s by a unitriangular matrix:
and
Thus \(x=x_0\cdots x_{n-1}\) is equal to \(y=y_0\cdots y_{n-1}\).
We define pure spinors \(x_I\) analogously, by
Our goal is to compute the expansion of \(x_I\) in the basis \(y_K\).
In fact, we will be free to multiply by the inverse of the matrix \(\overline{C}=(\overline{c}_{ji})\) and assume that \(x_{\overline{\imath }}\) is written
In this case, the fact that the \(x_{\overline{\imath }}\) span an isotropic space is equivalent to the fact that the matrix \(A=(a_{ji})\) is skew-symmetric. Up to appropriately indexing rows and columns, the x basis is related to the y basis by a matrix of the form
where \(w_\circ \) is the matrix with 1’s on the antidiagonal and zeroes elsewhere. The top n rows determine a skew-symmetric matrix (by replacing \(w_\circ \) with \(-B^t\)). For a subset \(I\subseteq [n]\), let B(I) be the submatrix formed by taking columns I of B, and let A(I) be the skew-symmetric matrix
Only the top n rows are needed to perform operations with such a matrix. For instance, if \(n=3\) and \(I=\{1\}\), the top rows are
and
For even r, the Pfaffian of any \(r\times r\) skew-symmetric matrix A may be computed recursively using the Laplace-type expansion formula
where \({{\,\mathrm{Pf}\,}}_{\widehat{j,r}}(A)\) is the Pfaffian of the submatrix of A obtained by removing the jth and rth rows and columns. Let \({{\,\mathrm{Pf}\,}}_K(A)\) denote the Pfaffian of the submatrix on rows and columns K, and we always use the convention \({{\,\mathrm{Pf}\,}}_\emptyset (A)=1\).
Theorem
For a subset \(I\subseteq [n]\), we have
where the sum is over subsets K of even cardinality.
See also [43, Corollary 2.4] for a related Pfaffian identity.
Proof
First we establish the formula for \(x_\emptyset \). In fact, for any \(m\le n\), we have
where \(K' = \{k'_1<\cdots <k'_s\} = [m]\smallsetminus K\). This is proved by induction on m, the base case \(m=0\) being the tautology \(y=y\). For the induction step, we compute using the Laplace expansion formula:
Note
So the second sum above becomes
where \(K'' = K'\smallsetminus j\). Combining the two and using Laplace expansion yields the claimed formula for \(x_\emptyset \).
The general case is similar. Using (A.3), it is equivalent to show
and we do this by induction on r. Let \(I=\{i_1<\cdots <i_r\}\) and consider \(i>i_r\). We compute:
as desired.
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Anderson, D., Nigro, A. (2020). Minuscule Schubert Calculus and the Geometric Satake Correspondence. In: Hu, J., Li, C., Mihalcea, L.C. (eds) Schubert Calculus and Its Applications in Combinatorics and Representation Theory. ICTSC 2017. Springer Proceedings in Mathematics & Statistics, vol 332. Springer, Singapore. https://doi.org/10.1007/978-981-15-7451-1_6
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