Consider the complex Lie group $SO(10)$ of type $D_5$. We denote $V(\omega)$ the highest weight representation of $SO(10)$ with highest weight $\omega$. We denote by $\omega_1,...,\omega_5$ the fundamental weights in the root system of type $D_5$.
We will obtain the minimal free resolution of the coordinate ring of the spinor variety of type $D_5$ (see Rincon - Isotropical linear spaces and valuated delta-matroids, Sec. 2, for a concise introduction to spinor varieties). The affine cone over the spinor variety of type $D_5$ lives in the representation $V(\omega_5)$, the 5th fundamental representation of $SO(10)$, considered as an affine space. Polynomial functions on this affine space are given by the symmetric algebra over the dual representation, i.e., $V(\omega_4)$.
Following the description in Fulton, Harris - Representation Theory, Ch. 20.1, we can construct $V(\omega_4)$ as $\wedge^0 E \oplus \wedge^2 E \oplus \wedge^4 E$, where $E$ is a 5 dimensional complex vector space. Let $\{e_0,...,e_4\}$ be a basis of $E$. Then a basis of $V(\omega_4)$ is given by the exterior products $e_J = e_{j_1} \wedge ... \wedge e_{j_{2r}}$, for all subsets $J=\{j_1,...,j_{2r}\}$ of even cardinality of $\{0,..., 4\}$. Denote by $x_J$ the variable corresponding to $e_J$ in $R$.
The spinor variety of type $D_5$ is cut out by quadratic equations which represent all possible relations among the sub Pfaffians of a $5\times 5$ generic skew symmetric matrix. A general description can be found for example in Manivel - On Spinor Varieties and Their Secants.
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The root system of type $D_5$ is contained in $\RR^5$. It is easy to express the weight of each variable of the ring $R$ with respect to the coordinate basis of $\RR^5$. The weight of $x_J$ is a vector $(a_1,...,a_5)\in\RR^5$, with $a_k = 1/2$ if $k\in J$ and $a_k = -1/2$ otherwise.
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Now we convert these weights into the basis of fundamental weights. To achieve this we make each previous weight into a column vector and join all column vectors into a matrix. Then we multiply on the left by the matrix $M$ expressing the change of basis from the coordinate basis of $\RR^5$ to the base of simple roots of $D_5$ (as described in Humphreys - Introduction to Lie Algebras and Representation Theory, Ch. 12.1). Finally we multiply the resulting matrix on the left by $N$, the transpose of the Cartan matrix of $D_5$, which expresses the change of basis from the simple roots to the fundamental weights of $D_5$. The columns of the matrix thus obtained are the desired weights, so they can be attached to the ring $R$.
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At this stage, we can issue the command to decompose the resolution.
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We deduce that the resolution has the following structure $$R \leftarrow V(\omega_1) \otimes R(-2) \leftarrow V(\omega_5) \otimes R(-3) \leftarrow V(\omega_4) \otimes R(-5) \leftarrow V(\omega_1) \otimes R(-6) \leftarrow R(-8) \leftarrow 0$$
Let us also decompose some graded components of the quotient $R/I$.
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We deduce that, for $d\in\{0,...,4\}$, $(R/I)_d = V(d\omega_4)$.