the following formulas, where $I$ is the identity matrix, $a \in \mathbb{C}$, and $i, j$ are elements in $\{1, 2, 3, 4\}$ such that $i < j$:

$$x_{(\varepsilon_i - \varepsilon_j)^\vee}(a) = I + a(E_{i,j} - E_{9-j,9-i}), \quad x_{(\varepsilon_i + \varepsilon_j)^\vee}(a) = I + a(E_{i,9-j} - E_{j,9-i}),$$ $$x_{(\varepsilon_j - \varepsilon_i)^\vee}(a) = I + a(E_{9-i,9-j} - E_{j,i}), \quad x_{(-\varepsilon_i - \varepsilon_j)^\vee}(a) = I + a(E_{9-i,j} - E_{9-j,i}).$$

For each root $\alpha$, we define a map $\chi_{\alpha}: \mathbb{C}^{10} \to G^{\vee}(\mathbb{C}[z, z^{-1}])$ by the formula

$$\chi_{\alpha}(\mathbf{a})(z) = \left( \prod_{k=1}^{8} x_{\beta_k^{\vee}}(a_k) \right) x_{\alpha^{\vee}}(a_9 + z a_{10}) z^{\alpha}$$

where $\mathbf{a}$ stands for the tuple $(a_1, \dots, a_{10}) \in \mathbb{C}^{10}$ and where $\beta_1^{\vee}, \dots, \beta_8^{\vee}$ are the coroots $\beta^{\vee}$ such that $\langle \beta^{\vee}, \alpha \rangle = 1$. We specify the enumeration in our cases of interest as follows.

<table><thead><tr><th>α</th><th>β<sub>1</sub><sup>√</sup></th><th>β<sub>2</sub><sup>√</sup></th><th>β<sub>3</sub><sup>√</sup></th><th>β<sub>4</sub><sup>√</sup></th><th>β<sub>5</sub><sup>√</sup></th><th>β<sub>6</sub><sup>√</sup></th><th>β<sub>7</sub><sup>√</sup></th><th>β<sub>8</sub><sup>√</sup></th></tr></thead><tbody><tr><th>ε<sub>1</sub>+ε<sub>2</sub></th><td>(ε<sub>1</sub>-ε<sub>3</sub>)<sup>√</sup></td><td>(ε<sub>1</sub>-ε<sub>4</sub>)<sup>√</sup></td><td>(ε<sub>1</sub>+ε<sub>4</sub>)<sup>√</sup></td><td>(ε<sub>1</sub>+ε<sub>3</sub>)<sup>√</sup></td><td>(ε<sub>2</sub>-ε<sub>3</sub>)<sup>√</sup></td><td>(ε<sub>2</sub>-ε<sub>4</sub>)<sup>√</sup></td><td>(ε<sub>2</sub>+ε<sub>4</sub>)<sup>√</sup></td><td>(ε<sub>2</sub>+ε<sub>3</sub>)<sup>√</sup></td></tr><tr><th>-ε<sub>1</sub>-ε<sub>2</sub></th><td>(-ε<sub>2</sub>-ε<sub>3</sub>)<sup>√</sup></td><td>(-ε<sub>2</sub>-ε<sub>4</sub>)<sup>√</sup></td><td>(ε<sub>4</sub>-ε<sub>2</sub>)<sup>√</sup></td><td>(ε<sub>3</sub>-ε<sub>2</sub>)<sup>√</sup></td><td>(-ε<sub>1</sub>-ε<sub>3</sub>)<sup>√</sup></td><td>(-ε<sub>1</sub>-ε<sub>4</sub>)<sup>√</sup></td><td>(ε<sub>4</sub>-ε<sub>1</sub>)<sup>√</sup></td><td>(ε<sub>3</sub>-ε<sub>1</sub>)<sup>√</sup></td></tr><tr><th>ε<sub>2</sub>-ε<sub>3</sub></th><td>(ε<sub>2</sub>-ε<sub>1</sub>)<sup>√</sup></td><td>(ε<sub>2</sub>-ε<sub>4</sub>)<sup>√</sup></td><td>(ε<sub>2</sub>+ε<sub>4</sub>)<sup>√</sup></td><td>(ε<sub>1</sub>+ε<sub>2</sub>)<sup>√</sup></td><td>(-ε<sub>1</sub>-ε<sub>3</sub>)<sup>√</sup></td><td>(-ε<sub>3</sub>-ε<sub>4</sub>)<sup>√</sup></td><td>(ε<sub>4</sub>-ε<sub>3</sub>)<sup>√</sup></td><td>(ε<sub>1</sub>-ε<sub>3</sub>)<sup>√</sup></td></tr><tr><th>ε<sub>3</sub>-ε<sub>2</sub></th><td>(ε<sub>3</sub>-ε<sub>1</sub>)<sup>√</sup></td><td>(ε<sub>3</sub>-ε<sub>4</sub>)<sup>√</sup></td><td>(ε<sub>3</sub>+ε<sub>4</sub>)<sup>√</sup></td><td>(ε<sub>1</sub>+ε<sub>3</sub>)<sup>√</sup></td><td>(-ε<sub>1</sub>-ε<sub>2</sub>)<sup>√</sup></td><td>(-ε<sub>2</sub>-ε<sub>4</sub>)<sup>√</sup></td><td>(ε<sub>4</sub>-ε<sub>2</sub>)<sup>√</sup></td><td>(ε<sub>1</sub>-ε<sub>2</sub>)<sup>√</sup></td></tr></tbody></table>

We now define two charts on $Gr_2^\lambda$, both with $\mathbb{C}^{22}$ as domain:

$$\phi_1 : (x_1, x_2, \mathbf{a}, \mathbf{b}) \mapsto (x_1, x_2; [\chi_{\varepsilon_1+\varepsilon_2}(\mathbf{a})(z-x_1), \chi_{-\varepsilon_1-\varepsilon_2}(\mathbf{b})(z-x_2)]),$$

$$\phi_2 : (x_1, x_2, \mathbf{a}', \mathbf{b}') \mapsto (x_1, x_2; [\chi_{\varepsilon_2-\varepsilon_2}(\mathbf{a}')(z-x_1), \chi_{\varepsilon_3-\varepsilon_2}(\mathbf{b}')(z-x_2)]).$$

One can then compute the transition map between these two charts. (The calculations were actually carried out with SINGULAR [11].) One finds the variables $a'_1, \dots, b'_{10}$ as rational functions in $x_2 - x_1, a_1, \dots, b_{10}$. We denote by $f$ the l.c.m. of the denominators.

Recall the notation used in Sect. 5.4. In the chart $\phi_1$, the cycle $Y(\frac{1}{2}, \frac{2}{1})$ is defined by the equations $a_1 = \dots = a_{10} = x_2 - x_1 = 0$, so the ideal in $R = \mathbb{C}[x_1, x_2, a_1, \dots, a_{10}, b_1, \dots, b_{10}]$ of

$$V = \phi_1^{-1}(Y(\frac{1}{2}, \frac{2}{1}))$$

is

$$p = (a_1, \dots, a_{10}, x_2 - x_1).$$

In the chart $\phi_2$, the cycle $X(\frac{2}{3}, \frac{3}{2})$ is defined by the equations $a'_2 = a'_3 = a'_4 = a'_8 = a'_9 = a'_{10} = b'_2 = b'_3 = b'_4 = b'_8 = 0$. Since the zero locus of $f$ contains the locus where the transition map between the charts is not defined, the ideal $\mathfrak{a}$ of the subvariety

$$X = \phi_1^{-1}(X(\frac{2}{3}, \frac{3}{2}))$$

is the preimage in $R$ of the ideal $\mathfrak{a}_f = (a'_2, a'_3, a'_4, a'_8, a'_9, a'_{10}, b'_2, b'_3, b'_4, b'_8)$ of the localized ring $R_f$. SINGULAR gives the following expression:
the following formulas, where $I$ is the identity matrix, $a \in \mathbb{C}$, and $i, j$ are elements in $\{1, 2, 3, 4\}$ such that $i < j$:

$$x_{(\varepsilon_i - \varepsilon_j)^\vee}(a) = I + a(E_{i,j} - E_{9-j,9-i}), \quad x_{(\varepsilon_i + \varepsilon_j)^\vee}(a) = I + a(E_{i,9-j} - E_{j,9-i}),$$ $$x_{(\varepsilon_j - \varepsilon_i)^\vee}(a) = I + a(E_{9-i,9-j} - E_{j,i}), \quad x_{(-\varepsilon_i - \varepsilon_j)^\vee}(a) = I + a(E_{9-i,j} - E_{9-j,i}).$$

For each root $\alpha$, we define a map $\chi_{\alpha} : \mathbb{C}^{10} \to G^{\vee}(\mathbb{C}[z, z^{-1}])$ by the formula

$$\chi_{\alpha}(\mathbf{a})(z) = \left( \prod_{k=1}^{8} x_{\beta_k^{\vee}}(a_k) \right) x_{\alpha^{\vee}}(a_9 + z a_{10}) z^{\alpha}$$

where $\mathbf{a}$ stands for the tuple $(a_1, \dots, a_{10}) \in \mathbb{C}^{10}$ and where $\beta_1^{\vee}, \dots, \beta_8^{\vee}$ are the coroots $\beta^{\vee}$ such that $\langle \beta^{\vee}, \alpha \rangle = 1$. We specify the enumeration in our cases of interest as follows.

<table><thead><tr><th>α</th><th>β<sub>1</sub><sup>√</sup></th><th>β<sub>2</sub><sup>√</sup></th><th>β<sub>3</sub><sup>√</sup></th><th>β<sub>4</sub><sup>√</sup></th><th>β<sub>5</sub><sup>√</sup></th><th>β<sub>6</sub><sup>√</sup></th><th>β<sub>7</sub><sup>√</sup></th><th>β<sub>8</sub><sup>√</sup></th></tr></thead><tbody><tr><th>ε<sub>1</sub>+ε<sub>2</sub></th><td>(ε<sub>1</sub>-ε<sub>3</sub>)<sup>√</sup></td><td>(ε<sub>1</sub>-ε<sub>4</sub>)<sup>√</sup></td><td>(ε<sub>1</sub>+ε<sub>4</sub>)<sup>√</sup></td><td>(ε<sub>1</sub>+ε<sub>3</sub>)<sup>√</sup></td><td>(ε<sub>2</sub>-ε<sub>3</sub>)<sup>√</sup></td><td>(ε<sub>2</sub>-ε<sub>4</sub>)<sup>√</sup></td><td>(ε<sub>2</sub>+ε<sub>4</sub>)<sup>√</sup></td><td>(ε<sub>2</sub>+ε<sub>3</sub>)<sup>√</sup></td></tr><tr><th>-ε<sub>1</sub>-ε<sub>2</sub></th><td>(-ε<sub>2</sub>-ε<sub>3</sub>)<sup>√</sup></td><td>(-ε<sub>2</sub>-ε<sub>4</sub>)<sup>√</sup></td><td>(ε<sub>4</sub>-ε<sub>2</sub>)<sup>√</sup></td><td>(ε<sub>3</sub>-ε<sub>2</sub>)<sup>√</sup></td><td>(-ε<sub>1</sub>-ε<sub>3</sub>)<sup>√</sup></td><td>(-ε<sub>1</sub>-ε<sub>4</sub>)<sup>√</sup></td><td>(ε<sub>4</sub>-ε<sub>1</sub>)<sup>√</sup></td><td>(ε<sub>3</sub>-ε<sub>1</sub>)<sup>√</sup></td></tr><tr><th>ε<sub>2</sub>-ε<sub>3</sub></th><td>(ε<sub>2</sub>-ε<sub>1</sub>)<sup>√</sup></td><td>(ε<sub>2</sub>-ε<sub>4</sub>)<sup>√</sup></td><td>(ε<sub>2</sub>+ε<sub>4</sub>)<sup>√</sup></td><td>(ε<sub>1</sub>+ε<sub>2</sub>)<sup>√</sup></td><td>(-ε<sub>1</sub>-ε<sub>3</sub>)<sup>√</sup></td><td>(-ε<sub>3</sub>-ε<sub>4</sub>)<sup>√</sup></td><td>(ε<sub>4</sub>-ε<sub>3</sub>)<sup>√</sup></td><td>(ε<sub>1</sub>-ε<sub>3</sub>)<sup>√</sup></td></tr><tr><th>ε<sub>3</sub>-ε<sub>2</sub></th><td>(ε<sub>3</sub>-ε<sub>1</sub>)<sup>√</sup></td><td>(ε<sub>3</sub>-ε<sub>4</sub>)<sup>√</sup></td><td>(ε<sub>3</sub>+ε<sub>4</sub>)<sup>√</sup></td><td>(ε<sub>1</sub>+ε<sub>3</sub>)<sup>√</sup></td><td>(-ε<sub>1</sub>-ε<sub>2</sub>)<sup>√</sup></td><td>(-ε<sub>2</sub>-ε<sub>4</sub>)<sup>√</sup></td><td>(ε<sub>4</sub>-ε<sub>2</sub>)<sup>√</sup></td><td>(ε<sub>1</sub>-ε<sub>2</sub>)<sup>√</sup></td></tr></tbody></table>

We now define two charts on $Gr_2^\lambda$, both with $\mathbb{C}^{22}$ as domain:

$$\phi_1 : (x_1, x_2, \mathbf{a}, \mathbf{b}) \mapsto (x_1, x_2; [\chi_{\varepsilon_1+\varepsilon_2}(\mathbf{a})(z-x_1), \chi_{-\varepsilon_1-\varepsilon_2}(\mathbf{b})(z-x_2)]),$$

$$\phi_2 : (x_1, x_2, \mathbf{a}', \mathbf{b}') \mapsto (x_1, x_2; [\chi_{\varepsilon_2-\varepsilon_2}(\mathbf{a}')(z-x_1), \chi_{\varepsilon_3-\varepsilon_2}(\mathbf{b}')(z-x_2)]).$$

One can then compute the transition map between these two charts. (The calculations were actually carried out with SINGULAR [11].) One finds the variables $a'_1, \dots, b'_{10}$ as rational functions in $x_2 - x_1, a_1, \dots, b_{10}$. We denote by $f$ the l.c.m. of the denominators.

Recall the notation used in Sect. 5.4. In the chart $\phi_1$, the cycle $Y(\frac{1}{2}, \frac{2}{1})$ is defined by the equations $a_1 = \dots = a_{10} = x_2 - x_1 = 0$, so the ideal in $R = \mathbb{C}[x_1, x_2, a_1, \dots, a_{10}, b_1, \dots, b_{10}]$ of

$$V = \phi_1^{-1}(Y(\frac{1}{2}, \frac{2}{1}))$$

is

$$p = (a_1, \dots, a_{10}, x_2 - x_1).$$

In the chart $\phi_2$, the cycle $X(\frac{2}{3}, \frac{3}{2})$ is defined by the equations $a'_2 = a'_3 = a'_4 = a'_8 = a'_9 = a'_{10} = b'_2 = b'_3 = b'_4 = b'_8 = 0$. Since the zero locus of $f$ contains the locus where the transition map between the charts is not defined, the ideal $\mathfrak{q}$ of the subvariety

$$X = \phi_1^{-1}(X(\frac{2}{3}, \frac{3}{2}))$$

is the preimage in $R$ of the ideal $\mathfrak{q}_f = (a'_2, a'_3, a'_4, a'_8, a'_9, a'_{10}, b'_2, b'_3, b'_4, b'_8)$ of the localized ring $R_f$. SINGULAR gives the following expression:
$$ q = (a_1a_4 + a_2a_3, a_1a_6 - a_2a_5, a_3a_6 + a_4a_5, a_1a_7 - a_3a_5, a_2a_7 + a_4a_5, a_1a_8 - a_4a_5, \\ a_2a_8 - a_4a_6, a_3a_8 - a_4a_7, a_5a_8 + a_6a_7, a_9, a_{10}, a_1b_4 + a_2b_3 + a_3b_2 + a_4b_1, \\ a_5b_4 + a_6b_3 + a_7b_2 + a_8b_1, a_3b_6 + a_4b_5 + a_7b_2 + a_8b_1 - (x_2-x_1), a_2b_7 - a_3b_6 + a_6b_3 - a_7b_2, \\ a_1b_8 + a_3b_6 + a_5b_4 + a_7b_2, a_2b_8 - a_4b_6 + a_6b_4 - a_8b_2, a_3b_8 - a_4b_7 + a_7b_4 - a_8b_3). $$

We observe that $q \subseteq p$, hence $V \subseteq X$.

Let $a$ and $x$ be two indeterminates. Let $B$ be the field $C(x, b_1, \dots, b_{10})$. Extract the last seven equations from $q$ and remove the term $x_2 - x_1$ in the third one; we then deal with seven linear equations with coefficients in $B$ in the eight variables $a_1, \dots, a_8$. This system has a nonzero solution $(c_1, \dots, c_8) \in B^8$. We can then define an algebra morphism $u : R/q \to B[a]/(a^2)$ by

$$ u(x_1) = u(x_2) = x, \quad u(b_i) = b_i \quad \text{for } i \in \{1, \dots, 10\}, $$

$$ u(a_9) = u(a_{10}) = 0, \quad u(a_i) = c_i a \quad \text{for } i \in \{1, \dots, 8\}. $$

The ring $B[a]/(a^2)$ is local with maximal ideal $(a)$ and the preimage of this ideal by $u$ is the ideal $p/q$ of $R/q$.

Let $A$ be the localization of $R/q$ at $p/q$. Then $u$ extends to an algebra morphism $\bar{u} : A \to B[a]/(a^2)$. By construction, the kernel of $\bar{u}$ contains $x_2 - x_1$ but not all of $a_1, \dots, a_8$. Therefore $x_2 - x_1$ does not generate the maximal ideal of $A$. Since $A$ is the local ring $\mathcal{O}_{V,X}$ of $X$ along $V$, this means that the order of vanishing of $x_2 - x_1$ along $V$ is larger than 1. In other words, the multiplicity of $\mathcal{Y}(\frac{1}{2}, \frac{2}{1})$ in the intersection product $\mathcal{X}(\frac{2}{3}, \frac{3}{2}) \cdot \mathcal{G}r_2^\lambda|_\Delta$ is larger than 1. Applying Theorem 5.8, we conclude that in (35) the coefficient of $(\frac{1}{2}, \frac{2}{1})$ is strictly larger than 1.

*Acknowledgements.* P.B. thanks Joel Kamnitzer for several discussions about the fusion product, which led to the main result of Sect. 5, and gratefully acknowledges the hospitality of the Centre de Recherches Mathématiques in Montreal, where research reported in Sect. 7.1 was carried out. The authors warmly thank Simon Riche for help with several technicalities. Last but not least, they thank an anonymous referee for his/her very careful reading and his/her numerous corrections and suggestions that improved the writing of this paper.

*Funding.* P.B. and S.G.'s research is in part supported by the ANR project GeoLie, ANR-15-CE40-0012. P.L.'s research is in part supported by the SFB/TRR 191 'Symplectic Structures in Geometry, Algebra and Dynamics'.

## References

[1] Baumann, P., Demarais, A.: Mirković-Vilonen basis in type $A_1$. Represent. Theory 25, 780–806 (2021) Zbl 7412478 MR 4319510

[2] Baumann, P., Gaussent, S.: On Mirković-Vilonen cycles and crystal combinatorics. Represent. Theory 12, 83–130 (2008) Zbl 1217.20028 MR 2390669

[3] Baumann, P., Kamnitzer, J., Knutson, A.: The Mirković-Vilonen basis and Duistermaat-Heckman measures. Acta Math. 227, 1–101 (2021) Zbl 07454649 MR 4346265
$$ q = (a_1a_4 + a_2a_3, a_1a_6 - a_2a_5, a_3a_6 + a_4a_5, a_1a_7 - a_3a_5, a_2a_7 + a_4a_5, a_1a_8 - a_4a_5, \\ a_2a_8 - a_4a_6, a_3a_8 - a_4a_7, a_5a_8 + a_6a_7, a_9, a_{10}, a_1b_4 + a_2b_3 + a_3b_2 + a_4b_1, \\ a_5b_4 + a_6b_3 + a_7b_2 + a_8b_1, a_3b_6 + a_4b_5 + a_7b_2 + a_8b_1 - (x_2-x_1), a_2b_7 - a_3b_6 + a_6b_3 - a_7b_2, \\ a_1b_8 + a_3b_6 + a_5b_4 + a_7b_2, a_2b_8 - a_4b_6 + a_6b_4 - a_8b_2, a_3b_8 - a_4b_7 + a_7b_4 - a_8b_3). $$

We observe that $q \subseteq p$, hence $V \subseteq X$.

Let $a$ and $x$ be two indeterminates. Let $B$ be the field $C(x, b_1, \dots, b_{10})$. Extract the last seven equations from $q$ and remove the term $x_2 - x_1$ in the third one; we then deal with seven linear equations with coefficients in $B$ in the eight variables $a_1, \dots, a_8$. This system has a nonzero solution $(c_1, \dots, c_8) \in B^8$. We can then define an algebra morphism $u : R/q \to B[a]/(a^2)$ by

$$ u(x_1) = u(x_2) = x, \quad u(b_i) = b_i \quad \text{for } i \in \{1, \dots, 10\}, $$

$$ u(a_9) = u(a_{10}) = 0, \quad u(a_i) = c_i a \quad \text{for } i \in \{1, \dots, 8\}. $$

The ring $B[a]/(a^2)$ is local with maximal ideal $(a)$ and the preimage of this ideal by $u$ is the ideal $p/q$ of $R/q$.

Let $A$ be the localization of $R/q$ at $p/q$. Then $u$ extends to an algebra morphism $\bar{u} : A \to B[a]/(a^2)$. By construction, the kernel of $\bar{u}$ contains $x_2 - x_1$ but not all of $a_1, \dots, a_8$. Therefore $x_2 - x_1$ does not generate the maximal ideal of $A$. Since $A$ is the local ring $\mathcal{O}_{V,X}$ of $X$ along $V$, this means that the order of vanishing of $x_2 - x_1$ along $V$ is larger than 1. In other words, the multiplicity of $\mathcal{Y}(\frac{1}{2}, \frac{2}{1})$ in the intersection product $\mathcal{X}(\frac{2}{3}, \frac{3}{2}) \cdot \mathcal{G}r_2^\lambda|_\Delta$ is larger than 1. Applying Theorem 5.8, we conclude that in (35) the coefficient of $(\frac{1}{2}, \frac{2}{1})$ is strictly larger than 1.

*Acknowledgements.* P.B. thanks Joel Kamnitzer for several discussions about the fusion product, which led to the main result of Sect. 5, and gratefully acknowledges the hospitality of the Centre de Recherches Mathématiques in Montreal, where research reported in Sect. 7.1 was carried out. The authors warmly thank Simon Riche for help with several technicalities. Last but not least, they thank an anonymous referee for his/her very careful reading and his/her numerous corrections and suggestions that improved the writing of this paper.

*Funding.* P.B. and S.G.'s research is in part supported by the ANR project GeoLie, ANR-15-CE40-0012. P.L.'s research is in part supported by the SFB/TRR 191 'Symplectic Structures in Geometry, Algebra and Dynamics'.

## References

[1] Baumann, P., Demarais, A.: Mirković-Vilonen basis in type $A_1$. Represent. Theory 25, 780–806 (2021) Zbl 7412478 MR 4319510

[2] Baumann, P., Gaussent, S.: On Mirković-Vilonen cycles and crystal combinatorics. Represent. Theory 12, 83–130 (2008) Zbl 1217.20028 MR 2390669

[3] Baumann, P., Kamnitzer, J., Knutson, A.: The Mirković-Vilonen basis and Duistermaat-Heckman measures. Acta Math. 227, 1–101 (2021) Zbl 07454649 MR 4346265
[4] Baumann, P., Riche, S.: Notes on the geometric Satake equivalence. In: Relative Aspects in Representation Theory, Langlands Functionality and Automorphic Forms, Lecture Notes in Math. 2221, Springer, Cham, 1-134 (2018) Zbl 1450.22009 MR 3839695

[5] Beilinson, A. A., Bernstein, J., Deligne, P.: Faisceaux pervers. In: Analysis and Topology on Singular Spaces, I (Luminy, 1981), Astérisque **100**, 5-171 (1982) Zbl 0536.14011 MR 751966

[6] Beilinson, A., Drinfeld, V.: Quantization of Hitchin's integrable system and Hecke eigen-sheaves. http://www.math.uchicago.edu/~mitya/langlands.html

[7] Berenstein, A., Fomin, S., Zelevinsky, A.: Cluster algebras. III. Upper bounds and double Bruhat cells. Duke Math. J. **126**, 1-52 (2005) Zbl 1135.16013 MR 2110627

[8] Berenstein, A., Kazhdan, D.: Geometric and unipotent crystals. II. From unipotent bicrystals to crystal bases. In: Quantum Groups, Contemp. Math. 433, Amer. Math. Soc., Providence, RI, 13-88 (2007) Zbl 1154.14035 MR 2349617

[9] Braverman, A., Gaitsgory, D.: Crystals via the affine Grassmannian. Duke Math. J. **107**, 561-575 (2001) Zbl 1015.20030 MR 1828302

[10] Braverman, A., Gaitsgory, D.: Geometric Eisenstein series. Invent. Math. **150**, 287-384 (2002) Zbl 1046.11048 MR 1933587

[11] Decker, W., Greuel, G.-M., Pfister, G., Schönemann, H.: SINGULAR 4.0.2 — A computer algebra system for polynomial computations. http://www.singular.uni-kl.de (2015)

[12] Fontaine, B., Kamnitzer, J.: Cyclic sieving, rotation, and geometric representation theory. Selecta Math. (N.S.) **20**, 609-625 (2014) Zbl 1295.22018 MR 3177928

[13] Fontaine, B., Kamnitzer, J., Kuperberg, G.: Buildings, spiders, and geometric Satake. Compos. Math. **149**, 1871-1912 (2013) Zbl 1304.22016 MR 3133297

[14] Frenkel, I. B., Khovanov, M. G.: Canonical bases in tensor products and graphical calculus for $U_q(sI_2)$. Duke Math. J. **87**, 409-480 (1997) Zbl 0883.17013 MR 1446615

[15] Fulton, W.: Intersection Theory. 2nd ed., Ergeb. Math. Grenzgeb. (3), Springer, Berlin (1998) Zbl 0885.14002 MR 1644323

[16] Gaitsgory, D.: Construction of central elements in the affine Hecke algebra via nearby cycles. Invent. Math. **144**, 253-280 (2001) Zbl 1072.14055 MR 1826370

[17] Gaussent, S., Littelmann, P.: LS galleries, the path model, and MV cycles. Duke Math. J. **127**, 35-88 (2005) Zbl 1078.22007 MR 2126496

[18] Gaussent, S., Littelmann, P., Nguyen, A. H.: Knuth relations, tableaux and MV-cycles. J. Ramanujan Math. Soc. **28A**, 191-219 (2013) Zbl 1364.14039 MR 3115193

[19] Geiss, C., Leclerc, B., Schröer, J.: Semicanonical bases and preprojective algebras. Ann. Sci. École Norm. Sup. (4) **38**, 193-253 (2005) Zbl 1131.17006 MR 2144987

[20] Geiß, C., Leclerc, B., Schröer, J.: Rigid modules over preprojective algebras. Invent. Math. **165**, 589-632 (2006) Zbl 1167.16009 MR 2242628

[21] Geiss, C., Leclerc, B., Schröer, J.: Auslander algebras and initial seeds for cluster algebras. J. London Math. Soc. (2) **75**, 718-740 (2007) Zbl 1135.17007 MR 2352732

[22] Goncharov, A., Shen, L.: Geometry of canonical bases and mirror symmetry. Invent. Math. **202**, 487-633 (2015) Zbl 1355.14030 MR 3418241

[23] Henriques, A., Kamnitzer, J.: Crystals and coboundary categories. Duke Math. J. **132**, 191-216 (2006) Zbl 1123.22007 MR 2219257

[24] Iversen, B.: Cohomology of Sheaves. Universitext, Springer, Berlin (1986) Zbl 1272.55001 MR 842190

[25] Kang, S.-J., Kashiwara, M., Kim, M., Oh, S.-j.: Monoidal categorification of cluster algebras. J. Amer. Math. Soc. **31**, 349-426 (2018) Zbl 1460.13039 MR 3758148

[26] Kashiwara, M.: The crystal base and Littelmann's refined Demazure character formula. Duke Math. J. **71**, 839-858 (1993) Zbl 0794.17008 MR 1240605
[4] Baumann, P., Riche, S.: Notes on the geometric Satake equivalence. In: Relative Aspects in Representation Theory, Langlands Functionality and Automorphic Forms, Lecture Notes in Math. 2221, Springer, Cham, 1-134 (2018) Zbl 1450.22009 MR 3839695

[5] Beilinson, A. A., Bernstein, J., Deligne, P.: Faisceaux pervers. In: Analysis and Topology on Singular Spaces, I (Luminy, 1981), Astérisque **100**, 5-171 (1982) Zbl 0536.14011 MR 751966

[6] Beilinson, A., Drinfeld, V.: Quantization of Hitchin's integrable system and Hecke eigen-sheaves. http://www.math.uchicago.edu/~mitya/langlands.html

[7] Berenstein, A., Fomin, S., Zelevinsky, A.: Cluster algebras. III. Upper bounds and double Bruhat cells. Duke Math. J. **126**, 1-52 (2005) Zbl 1135.16013 MR 2110627

[8] Berenstein, A., Kazhdan, D.: Geometric and unipotent crystals. II. From unipotent bicrystals to crystal bases. In: Quantum Groups, Contemp. Math. 433, Amer. Math. Soc., Providence, RI, 13-88 (2007) Zbl 1154.14035 MR 2349617

[9] Braverman, A., Gaitsgory, D.: Crystals via the affine Grassmannian. Duke Math. J. **107**, 561-575 (2001) Zbl 1015.20030 MR 1828302

[10] Braverman, A., Gaitsgory, D.: Geometric Eisenstein series. Invent. Math. **150**, 287-384 (2002) Zbl 1046.11048 MR 1933587

[11] Decker, W., Greuel, G.-M., Pfister, G., Schönemann, H.: SINGULAR 4.0.2 — A computer algebra system for polynomial computations. http://www.singular.uni-kl.de (2015)

[12] Fontaine, B., Kamnitzer, J.: Cyclic sieving, rotation, and geometric representation theory. Selecta Math. (N.S.) **20**, 609-625 (2014) Zbl 1295.22018 MR 3177928

[13] Fontaine, B., Kamnitzer, J., Kuperberg, G.: Buildings, spiders, and geometric Satake. Compos. Math. **149**, 1871-1912 (2013) Zbl 1304.22016 MR 3133297

[14] Frenkel, I. B., Khovanov, M. G.: Canonical bases in tensor products and graphical calculus for $U_q(sI_2)$. Duke Math. J. **87**, 409-480 (1997) Zbl 0883.17013 MR 1446615

[15] Fulton, W.: Intersection Theory. 2nd ed., Ergeb. Math. Grenzgeb. (3), Springer, Berlin (1998) Zbl 0885.14002 MR 1644323

[16] Gaitsgory, D.: Construction of central elements in the affine Hecke algebra via nearby cycles. Invent. Math. **144**, 253-280 (2001) Zbl 1072.14055 MR 1826370

[17] Gaussent, S., Littelmann, P.: LS galleries, the path model, and MV cycles. Duke Math. J. **127**, 35-88 (2005) Zbl 1078.22007 MR 2126496

[18] Gaussent, S., Littelmann, P., Nguyen, A. H.: Knuth relations, tableaux and MV-cycles. J. Ramanujan Math. Soc. **28A**, 191-219 (2013) Zbl 1364.14039 MR 3115193

[19] Geiss, C., Leclerc, B., Schröer, J.: Semicanonical bases and preprojective algebras. Ann. Sci. École Norm. Sup. (4) **38**, 193-253 (2005) Zbl 1131.17006 MR 2144987

[20] Geiß, C., Leclerc, B., Schröer, J.: Rigid modules over preprojective algebras. Invent. Math. **165**, 589-632 (2006) Zbl 1167.16009 MR 2242628

[21] Geiss, C., Leclerc, B., Schröer, J.: Auslander algebras and initial seeds for cluster algebras. J. London Math. Soc. (2) **75**, 718-740 (2007) Zbl 1135.17007 MR 2352732

[22] Goncharov, A., Shen, L.: Geometry of canonical bases and mirror symmetry. Invent. Math. **202**, 487-633 (2015) Zbl 1355.14030 MR 3418241

[23] Henriques, A., Kamnitzer, J.: Crystals and coboundary categories. Duke Math. J. **132**, 191-216 (2006) Zbl 1123.22007 MR 2219257

[24] Iversen, B.: Cohomology of Sheaves. Universitext, Springer, Berlin (1986) Zbl 1272.55001 MR 842190

[25] Kang, S.-J., Kashiwara, M., Kim, M., Oh, S.-j.: Monoidal categorification of cluster algebras. J. Amer. Math. Soc. **31**, 349-426 (2018) Zbl 1460.13039 MR 3758148

[26] Kashiwara, M.: The crystal base and Littelmann's refined Demazure character formula. Duke Math. J. **71**, 839-858 (1993) Zbl 0794.17008 MR 1240605
[27] Kashiwara, M.: Global crystal bases of quantum groups. Duke Math. J. **69**, 455–485 (1993) Zbl 0774.17018 MR 1203234

[28] Kashiwara, M.: On crystal bases. In: Representations of Groups (Banff, AB, 1994), CMS Conf. Proc. 16, Amer. Math. Soc., Providence, RI, 155–197 (1995) Zbl 0851.17014 MR 1357199

[29] Kashiwara, M., Saito, Y.: Geometric construction of crystal bases. Duke Math. J. **89**, 9–36 (1997) Zbl 0901.17006 MR 1458969

[30] Khovanov, M., Kuperberg, G.: Web bases for sl(3) are not dual canonical. Pacific J. Math. **188**, 129–153 (1999) Zbl 0929.17012 MR 1680395

[31] Kumar, S.: Kac–Moody Groups, their Flag Varieties and Representation Theory. Progr. Math. 204, Birkhäuser Boston, Boston, MA (2002) Zbl 1026.17030 MR 1923198

[32] Lascoux, A., Schützenberger, M.-P.: Le monoïde plaxique. In: Noncommutative Structures in Algebra and Geometric Combinatorics (Naples, 1978), Quad. “Ricerca Sci.” 109, CNR, Rome, 129–156 (1981) Zbl 0517.20036 MR 646486

[33] Littelmann, P.: Paths and root operators in representation theory. Ann. of Math. (2) **142**, 499–525 (1995) Zbl 0858.17023 MR 1356780

[34] Littelmann, P.: A plactic algebra for semisimple Lie algebras. Adv. Math. **124**, 312–331 (1996) Zbl 0892.17009 MR 1424313

[35] Littelmann, P.: Characters of representations and paths in $\mathfrak{so}_R^*$. In: Representation Theory and Automorphic Forms (Edinburgh, 1996), Proc. Sympos. Pure Math. 61, Amer. Math. Soc., Providence, RI, 29–49 (1997) Zbl 0892.17010 MR 1476490

[36] Lusztig, G.: Singularities, character formulas, and a $q$-analog of weight multiplicities. In: Analysis and Topology on Singular Spaces, II, III (Luminy, 1981), Astérisque **101**, 208–229 (1983) Zbl 0561.22013 MR 737932

[37] Lusztig, G.: Introduction to Quantum Groups. Progr. Math. 110, Birkhäuser Boston, Boston, MA (1993) Zbl 0788.17010 MR 1227098

[38] Lysenko, S.: Moduli of metaplectic bundles on curves and theta-sheaves. Ann. Sci. École Norm. Sup. (4) **39**, 415–466 (2006) Zbl 1111.14029 MR 2265675

[39] Mirković, I., Vilonen, K.: Geometric Langlands duality and representations of algebraic groups over commutative rings. Ann. of Math. (2) **166**, 95–143 (2007) Zbl 1138.22013 MR 2342692

[40] Ngô, B. C., Polo, P.: Résolutions de Demazure affines et formule de Casselman–Shalika géométrique. J. Algebraic Geom. **10**, 515–547 (2001) Zbl 1041.14002 MR 1832331

[41] Pappas, G., Rapoport, M.: Twisted loop groups and their affine flag varieties. Adv. Math. **219**, 118–198 (2008) Zbl 1159.22010 MR 2435422

[42] Reich, R. C.: Twisted geometric Satake equivalence via gerbes on the factorizable Grassmannian. Represent. Theory **16**, 345–449 (2012) Zbl 1302.22014 MR 2956088

[43] Richarz, T.: A new approach to the geometric Satake equivalence. Doc. Math. **19**, 209–246 (2014) Zbl 1326.14121 MR 3178249

[44] Torres, J.: The symplectic plactic monoid, crystals, and MV cycles. Pacific J. Math. **286**, 439–497 (2017) Zbl 1354.05143 MR 3599867

[45] Xiao, L., Zhu, X.: Cycles on Shimura varieties via geometric Satake. arXiv:1707.05700 (2017)

[46] Zhu, X.: An introduction to affine Grassmannians and the geometric Satake equivalence. In: Geometry of Moduli Spaces and Representation Theory, IAS/Park City Math. Ser. 24, Amer. Math. Soc., Providence, RI, 59–154 (2017) Zbl 1453.14122 MR 3752460
[27] Kashiwara, M.: Global crystal bases of quantum groups. Duke Math. J. 69, 455–485 (1993) Zbl 0774.17018 MR 1203234

[28] Kashiwara, M.: On crystal bases. In: Representations of Groups (Banff, AB, 1994), CMS Conf. Proc. 16, Amer. Math. Soc., Providence, RI, 155–197 (1995) Zbl 0851.17014 MR 1357199

[29] Kashiwara, M., Saito, Y.: Geometric construction of crystal bases. Duke Math. J. 89, 9–36 (1997) Zbl 0901.17006 MR 1458969

[30] Khovanov, M., Kuperberg, G.: Web bases for sl(3) are not dual canonical. Pacific J. Math. 188, 129–153 (1999) Zbl 0929.17012 MR 1680395

[31] Kumar, S.: Kac–Moody Groups, their Flag Varieties and Representation Theory. Progr. Math. 204, Birkhäuser Boston, Boston, MA (2002) Zbl 1026.17030 MR 1923198

[32] Lascoux, A., Schützenberger, M.-P.: Le monoïde plaxique. In: Noncommutative Structures in Algebra and Geometric Combinatorics (Naples, 1978), Quad. “Ricerca Sci.” 109, CNR, Rome, 129–156 (1981) Zbl 0517.20036 MR 646486

[33] Littelmann, P.: Paths and root operators in representation theory. Ann. of Math. (2) **142**, 499–525 (1995) Zbl 0858.17023 MR 1356780

[34] Littelmann, P.: A plactic algebra for semisimple Lie algebras. Adv. Math. **124**, 312–331 (1996) Zbl 0892.17009 MR 1424313

[35] Littelmann, P.: Characters of representations and paths in $\mathfrak{so}_R^*$. In: Representation Theory and Automorphic Forms (Edinburgh, 1996), Proc. Sympos. Pure Math. 61, Amer. Math. Soc., Providence, RI, 29–49 (1997) Zbl 0892.17010 MR 1476490

[36] Lusztig, G.: Singularities, character formulas, and a $q$-analog of weight multiplicities. In: Analysis and Topology on Singular Spaces, II, III (Luminy, 1981), Astérisque **101**, 208–229 (1983) Zbl 0561.22013 MR 737932

[37] Lusztig, G.: Introduction to Quantum Groups. Progr. Math. 110, Birkhäuser Boston, Boston, MA (1993) Zbl 0788.17010 MR 1227098

[38] Lysenko, S.: Moduli of metaplectic bundles on curves and theta-sheaves. Ann. Sci. École Norm. Sup. (4) **39**, 415–466 (2006) Zbl 1111.14029 MR 2265675

[39] Mirković, I., Vilonen, K.: Geometric Langlands duality and representations of algebraic groups over commutative rings. Ann. of Math. (2) **166**, 95–143 (2007) Zbl 1138.22013 MR 2342692

[40] Ngô, B. C., Polo, P.: Résolutions de Demazure affines et formule de Casselman–Shalika géométrique. J. Algebraic Geom. **10**, 515–547 (2001) Zbl 1041.14002 MR 1832331

[41] Pappas, G., Rapoport, M.: Twisted loop groups and their affine flag varieties. Adv. Math. **219**, 118–198 (2008) Zbl 1159.22010 MR 2435422

[42] Reich, R. C.: Twisted geometric Satake equivalence via gerbes on the factorizable Grassmannian. Represent. Theory **16**, 345–449 (2012) Zbl 1302.22014 MR 2956088

[43] Richarz, T.: A new approach to the geometric Satake equivalence. Doc. Math. **19**, 209–246 (2014) Zbl 1326.14121 MR 3178249

[44] Torres, J.: The symplectic plactic monoid, crystals, and MV cycles. Pacific J. Math. **286**, 439–497 (2017) Zbl 1354.05143 MR 3599867

[45] Xiao, L., Zhu, X.: Cycles on Shimura varieties via geometric Satake. arXiv:1707.05700 (2017)

[46] Zhu, X.: An introduction to affine Grassmannians and the geometric Satake equivalence. In: Geometry of Moduli Spaces and Representation Theory, IAS/Park City Math. Ser. 24, Amer. Math. Soc., Providence, RI, 59–154 (2017) Zbl 1453.14122 MR 3752460