Note on the decomposition of states

TL;DR

This paper presents a precise decomposition formula for the state polytope and Hilbert-Mumford index of reducible varieties, simplifying previous proofs through convex geometry and character decomposition.

Contribution

It introduces a new, simplified proof of the state polytope decomposition for reducible varieties using convex geometry and character decomposition.

Findings

Derived a sharp decomposition formula for state polytopes

Simplified the proof of state polytope decomposition

Enhanced understanding of the geometric structure of reducible varieties

Abstract

We derive a sharp decomposition formula for the state polytope of the Hilbert point and the Hilbert-Mumford index of reducible varieties by using the decomposition of characters and basic convex geometry. This proof captures the essence of the decomposition of the state polytopes in general, and considerably simplifies an earlier proof by the author and Jaekwang Kim which uses a careful analysis of initial ideals of reducible varieties.

Equations

$$[I{]}_m:[I_m\hookrightarrow S_m]\in {\text{Gr}}_{Q(m)}{S}_m\hookrightarrow \mathbb{P}\left(\stackrel{Q(m)}{\bigwedge }{S}_m\right).$$

$$[I{]}_m^{\ast }:[S_m\to S_m/I_m]\in {\text{Gr}}^{P(m)}{S}_m\hookrightarrow \mathbb{P}\left(\stackrel{P(m)}{\bigwedge }{S}_m^{\ast }\right).$$

$$X_i\subset \{x_0=\cdots =x_{n_{i-1}-1}=0,x_{n_i+1}=x_{n_i+2}=\cdots =x_n=0\}.$$

$${\mathcal{P}}_m(I_X)=\sum _{i=1}^{\ell }{\mathcal{P}}_m(I_{X_i}\cap k[x_{n_{i-1}},\cdots ,x_{n_i}])+\sum _{i=1}^{\ell -1}{\mathcal{P}}_m(T_i\cap k[x_{n_{i-1}},\dots ,x_n])$$

$$\mu([X]^{*}_{m},\rho)=\sum_{i=1}^{\ell}\mu([X_{i}]^{*}_{m},\rho_{i})-\sum_{i=1}^{\ell}\bigg{(}\frac{mP_{i}(m)}{n_{i}-n_{i-1}+1}\sum_{k=n_{i-1}}^{n_{i}}r_{k}\bigg{)}+\frac{mP(m)}{n+1}\sum_{i=0}^{n}r_{i}+m\sum_{i=1}^{\ell-1}r_{n_{i}}$$

$$\mu ([I_X{]}_m,\rho )=\mu ([\mathbf{in}{}_{{\prec }_{\rho }}{I}_X{]}_m,\rho )=-\sum _{x^{\alpha }\in (\mathbf{in}{}_{{\prec }_{\rho }}{I}_X{)}_m}{\text{wt}}_{\rho }(x^{\alpha }).$$

$$\begin{array}[]{ccl}\mu([I_{X}]_{m}^{*},\rho)&=&\max\{-\sum_{x^{\alpha}\in\mathcal{B}}\textrm{wt}_{\rho}(f^{\alpha})\,|\,\mbox{$\mathcal{B}$ a monomial basis of $(S/I_{X})_{m}$}\}\\ &=&-\sum_{x^{\alpha}\in S_{m}\setminus{\bf in}\,_{\prec_{\rho}}(I_{X})}\left(-\mathrm{wt}_{\rho}(x^{\alpha})\right)=-\sum_{x^{\alpha}\in({\bf in}\,_{\prec_{\rho}}(I_{X}))_{m}}\mathrm{wt}_{\rho}(x^{\alpha}).\end{array}$$

$$S^{m}V=\sum _{i=1}^{\nu }{S}^m{V}_i+\sum _{\begin{array}{c}{\sum }_j{m}_j=m\\ 0<m_j<m\end{array}}\underset{j=1}{\overset{\nu }{⨂}}{S}^{m_j}{V}_j.$$

$$W_{\nu +1}=\sum _{\begin{array}{c}{\sum }_j{m}_j=m\\ 0<m_j<m\end{array}}\underset{j=1}{\overset{\nu }{⨂}}{S}^{m_j}{V}_j.$$

$${\chi }_s({\pi }_j(t))=\{\begin{array}{ll}{\chi }_s(t),& M_j\le s\le M_j^{\prime }\\ 1,& \text{mbox}else\end{array}.$$

$${\Xi }_{\left[{\bigwedge }^{N}I\right]}(R)=\sum _{j=1}^{\nu }{\Xi }_{\left[{\bigwedge }^{N_j}{I}_j\right]}(R_j)+{\Xi }_{\left[{\bigwedge }^{N_{\nu +1}}{I}_{\nu +1}\right]}(R).$$

$${\xi }_j=\sum _{\chi \in X(R)}({\xi }_j{)}_{\chi }=\sum _{\chi \in X(R_j)}({\xi }_j{)}_{\chi }.$$

$$\stackrel{N}{\bigwedge }I\simeq \underset{j=1}{\overset{\nu +1}{⨂}}\left(\stackrel{N_j}{\bigwedge }{I}_j\right)$$

$$\xi =\sum _{\begin{array}{c}{\chi }^{(j)}\in X(R_j)\\ {\chi }^{\nu +1}\in X(R)\end{array}}\underset{j=1}{\overset{\nu +1}{⨂}}({\xi }_j{)}_{{\chi }^{(j)}}$$

$$\begin{array}[]{l}X_{1}\subset\{x_{\ell+1}=\cdots=x_{n}=0\}\\ X_{2}\subset\{x_{0}=\cdots=x_{\ell-1}=0\}.\end{array}$$

$$(I_X{)}_m=(I_X\cap W_1)+(I_X\cap W_2)+(I_X\cap W_3).$$

$$††$$

$$H_1=\{\mathbf{x}\in {\mathbb{R}}^n\left|\sum _{i=1}^{n_1}{x}_i=N_1,x_i=0\forall i>n_1\right\}$$

$$H_2=\{\mathbf{x}\in {\mathbb{R}}^n\left|\sum _{i=n_1}^n{x}_i=N_2,x_i=0\forall i<n_1\right\}.$$

$$v_1+w_1=\sum _{i,j}{\lambda }_{ij}(v_i+w_j).$$

$$\left(1-\sum _j{\lambda }_{1j}\right)v_1-\sum _{i\ne 1}{\lambda }_{ij}{v}_i=\sum _{j\ne 1}{\lambda }_{ij}{w}_j+\left(\sum _i{\lambda }_{i1}-1\right)w_1\in H_1\cap H_2$$

$$v_{1k}=\sum _{\begin{array}{c}i\ne 1\\ 1\le j\le s\end{array}}{\mu }_{ij}{v}_{ik}k\ne n_1$$

$$v_{1n_1}$$

$$=\sum _{i\ne 1}{\mu }_{ij}\left(N_1-\sum _{k=0}^{n_1-1}{v}_{ik}\right)=\sum _{i\ne 1}{\mu }_{ij}{v}_{i{n}_1}$$

$$\left\{\tau +\sum _{i=1}^{\ell }{v}_i|v_i\in {\mathcal{V}}_i\right\}.$$

$$H_1=\left\{\mathbf{x}\in {\mathbb{R}}^n|\sum _{i=0}^{n_1}{x}_i=Q_1(m),x_i=0,\forall i>n_1\right\}$$

$$H_2=\left\{\mathbf{x}\in {\mathbb{R}}^n|\sum _{i=n_1}^n{x}_i=Q_2(m),x_i=0,\forall i<n_1\right\}$$

$$\mu ([I_X{]}_m,\rho )=\max \{-⟨\chi ,\rho ⟩|\chi \in {\Xi }_{[I_X{]}_m}(R)\}$$

$$\begin{array}[]{lcr}\langle\chi,\rho\rangle&=&\langle\chi_{1},\rho_{1}\rangle+\langle\chi_{2},\rho_{2}\rangle+\langle\tau,\rho\rangle.\\ \end{array}$$

$$\sum _{x^{\alpha }\in {\left(\mathbf{in}{}_{{\prec }_{\rho }}{I}_X\right)}_m}{\chi }^{\alpha }$$

$$\begin{array}[]{cll}\langle\chi,\rho\rangle&=&\langle\chi,\rho^{\prime}\rangle+\langle\chi,(w,w,\dots,w)\rangle\\ &=&-\mu([I_{X}]_{m},\rho)+w\sum_{x^{\alpha}\in\left({\bf in}\,_{\prec_{\rho}}I_{X}\right)_{m}}(\sum_{i=0}^{n}\alpha_{i})\\ &=&-\mu([I_{X}]_{m},\rho)+wmQ(m).\end{array}$$

Full text

The first named author was supported by the Research Resettlement Fund for the new faculty of Seoul National University, and the following grants funded by the government of Korea: NRF grant 2011-0030044 (SRC-GAIA) and NRF grant NRF-2013R1A1A2010649.

1. Introduction & Preliminaries

In this article, we take a new look at the decomposition formula for state polytopes [HK] from a more general point of view. We shall work over an algebraically closed field $k$ of characteristic zero. Let $G$ be a linear algebraic group and $R$ be a maximal torus of it. Let $W$ be a rational representation of $G$ and $w\in W$ be a point. Recall that the state $\Xi_{w}(R)$ of $w$ with respect to $R$ is the set of the characters $\chi\in X(R)$ such that $w_{\chi}\neq 0$. Since $w=\sum_{\chi}w_{\chi}$ implies $cw=\sum_{\chi}cw_{\chi}$, we have $\Xi_{w}(R)=\Xi_{cw}(R)$ for any nonzero $c\in k$. Hence we may define the state $\Xi_{p}(R)$ of $p\in\mathbb{P}(W)$ to be $\Xi_{w}(R)$ for any affine point $w\in W$ over $p$. (We conflate a vector space $W$ with the affine scheme $\textup{Spec \,}\mathrm{Sym}(W^{*})$.)

We shall be concerned with the states of Hilbert points of homogeneous ideals. Let $S=k[x_{0},\dots,x_{n}]$ and let $P(u)$ be a rational polynomial in one variable $u$ and $Q(u)=\binom{u+m}{m}-P(u)$. If $m$ is at least the Gotzmann number of $P(u)$, then for any saturated homogeneous polynomial $I\subset k[x_{0},\dots,x_{n}]$ whose Hilbert polynomial is $P$, the $m$th Hilbert point $[I]_{m}$ of $I$ is well defined as a $Q(m)$-dimensional subspace of the degree $m$ piece $S_{m}$:

[TABLE]

The dual $m$th Hilbert point $[I]_{m}^{*}$ is defined by

[TABLE]

The collection of $m$th Hilbert points form a closed subscheme $\textup{Hilb}_{P}\mathbb{P}(V)$, called the Hilbert scheme, of the Grassmannian. Working out the geometric invariant theory (GIT) of suitable Hilbert schemes give rise to various moduli spaces [Mum65], and our main motivation for the study in this article is the construction of the moduli of curves. The link between GIT and the study of state polytopes is given by the following fundamental observation (numerical criterion): *If $G$ is reductive and $V$ is a rational representation, $v\in V$ is GIT unstable if and only if there is a torus $R$ of $G$ such that the convex hull of $\Xi_{v}(R)$ does not contain the trivial character. *

The monomial basis of $\bigwedge^{Q(m)}S_{m}$ consists of $x^{\alpha}:=x^{\alpha(1)}\wedge\cdots\wedge x^{\alpha(Q(m))}$, the wedge product of $Q(m)$ degree $m$ monomials $x^{\alpha(i)}$’s. The basis members are also the $R$-weight vectors of $\bigwedge^{Q(m)}S_{m}$, where $R$ is the maximal torus of $G={\mathrm{G}L}(S_{1})$ diagonalized by $x_{0},\dots,x_{n}$: Indeed, let $\chi_{i}$ be the character of $R$ determined by $t.x_{i}=\chi_{i}(t)x_{i}$. Then by letting $t_{i}$ denote $\chi_{i}(t)$ and using the usual multivector notation $t^{\gamma}=\prod_{i=0}^{n}t_{i}^{\gamma_{i}}$, $t\in T$, $\gamma\in\mathbb{Z}^{n+1}$, we have $t.x^{\alpha}=t^{\sum_{i=1}^{Q(m)}\alpha(i)}x^{\alpha}$, which means precisely that $\left(\bigwedge^{Q(m)}S_{m}\right)_{\chi^{\alpha}}=k.x^{\alpha}$ where $\chi^{\alpha}=\prod_{i=1}^{Q(m)}\prod_{j=0}^{n}\chi_{j}^{\alpha(i)_{j}}$.

The Hilbert point $[I]_{m}$ has a nonzero $x^{\alpha}$-coefficient if and only if the degree $m$ monomials other than $x^{\alpha(1)},\dots,x^{\alpha(Q(m))}$ form a $k$-basis of $S_{m}/I_{m}$. Following [Kem78], we denote the set of states by $\Xi_{[I]_{m}}(R)$ and its convex hull by $\mathcal{P}_{m}(I)$. We call $\mathcal{P}_{m}(I)$ the $m$th state polytope of $I$, following [BM88].

For any fixed sufficiently large $m$, Bayer and Morrison proved that

Theorem 1.1**.**

[BM88, Theorem 3.1]** There is a canonical bijection between the vertices of $\mathcal{P}_{m}(I)$ and the initial ideals ${\bf in}\,_{\prec}(I)$ as $\prec$ runs through all term orders on $k[x_{0},\dots,x_{n}]$.

Using the Bayer-Morrison theorem and basic properties of monomial orders and initial ideals, decomposition formulae for initial ideals, state polytopes, and Hilbert-Mumford indices were achieved in [HK]:

Theorem 1.2**.**

[HK]** Let $X$ be a chain of projective varieties $X_{1},\dots,X_{\ell}$ defined by a saturated homogeneous ideal $I_{X}=\cap_{i}I_{X_{i}}$ i.e. $X=\cup_{i=1}^{\ell}X_{i}$ and $X_{i}$ meets $X_{j}$ when and only when $|i-j|=1$. Suppose that there is a homogeneous coordinate system $x_{0},\dots,x_{n}$ and a sequence $n_{0}=0<n_{1}<\cdots<n_{\ell}=n$ such that

[TABLE]

Then the state polytope of $X$ is given by the following decomposition formula

[TABLE]

where $T_{i}=\langle x_{n_{i-2}},\dots,x_{n_{i-1}-1}\rangle\langle x_{n_{i}+1},\dots,x_{n}\rangle$ for $2\leq i\leq\ell-1$, and $T_{1}=\langle x_{n_{1}+1},x_{n_{1}+2},\dots,x_{n}\rangle$ and $T_{\ell}=\langle x_{n_{\ell-2}},x_{n_{\ell-2}+1},\dots,x_{n_{\ell-1}-1}\rangle$.

Here, ${\mathcal{P}}_{m}(I_{X_{i}}\cap k[x_{n_{i-1}},\cdots,x_{n_{i}}])$ is regarded as a convex polytope in the subspace $\{{\bf a}\in\mathbb{R}^{n+1}\,|\,a_{i}=0,\,\forall i<n_{i-1},\forall i>n_{i}\}.$ Similarly, ${\mathcal{P}}_{m}(T_{i}\cap k[x_{n_{i-1}},\dots,x_{n}])$ is also regarded as a convex polytope in the relevant vector subspace. Note that $\mathcal{P}_{m}(T_{i}\cap k[x_{n_{i-1}},\dots,x_{n}])$ is a point since $T_{i}$ is a monomial ideal. We let $\tau$ denote the point $\sum_{i=1}^{\ell-1}\mathcal{P}_{m}(T_{i}\cap k[x_{n_{i-1}},\dots,x_{n}])$. We shall show in Section 2.1 that Theorem 1.2 is a direct consequence of the fact that the characters of a direct sum is the sum of the characters (Proposition 2.1).

It is also shown in [HK] that this decomposition is sharp: the vertices of $\mathcal{P}_{m}(I_{X})$ are precisely the sums of vertices of $\mathcal{P}_{m}(I_{X_{i}}\cap k[x_{n_{i-1}},\cdots,x_{n_{i}}])$ and $\tau$ (Corollary 2.4). The proof in [HK] uses Theorem 1.1 and the initial ideal decomposition formula. We shall show in Section 2.2 that the sharpness of the decomposition is in fact a consequence of a general convex geometry phenomenon.

Finally, we also reprove in Section 3 the Hilbert-Mumford index decomposition formula below by using the decomposition of characters.

Proposition 1.3**.**

[HK]** Let $X$ be as in Theorem 1.2 and $\rho:\mathbb{G}_{m}\to{\mathrm{G}L}_{n+1}$ be a 1-parameter subgroup of ${\mathrm{G}L}_{n+1}$ diagonalized by $\{x_{0},\dots,x_{n}\}$ with weights $(r_{0},\cdots,r_{n})$ and $\rho_{i}$ be the restriction of $\rho$ to ${\mathrm{G}L}(kx_{n_{i-1}}+\cdots+kx_{n_{i}})$. Then the Hilbert-Mumford index $\mu([X]^{*}_{m},\rho)$ of the (dual) $m$th Hilbert point of $X$ with respect to $\rho$ is given by

[TABLE]

where $P(m)$ is the Hilbert polynomial of $I_{X}\subset k[x_{0},\dots,x_{n}]$ and $P_{i}(m)$, the Hilbert polynomial of $I_{X_{i}}\cap k[x_{n_{i-1}},\dots,x_{n_{i}}]$ regarded as an ideal in $k[x_{n_{i-1}},\dots,x_{n_{i}}]$.

We close this section with an observation that will be used in Section 3: We shall prove that the Hilbert-Mumford index of $[I_{X}]_{m}$ and that of the dual Hilbert point $[I_{X}]_{m}^{*}$ are the same. Let $\rho$ be a one-parameter subgroup of $SL(S_{1})$. We recall the fact that $\lim_{t\to 0}\rho(t).[I_{X}]_{m}=[{\bf in}\,_{\prec_{\rho}}I_{X}]_{m}$ where $\prec_{\rho}$ is the $\rho$-weight order with the reverse lexicographic tie-breaking [BM88]. Then the Hilbert-Mumford index is

[TABLE]

Let $\{f_{0},\dots,f_{n}\}\subset S_{1}^{*}$ be the dual basis of $\{x_{0},\dots,x_{n}\}\subset S_{1}$. Use the multi-vector notation $f^{\alpha}=\prod_{i=0}^{n}f_{i}^{\alpha_{i}}$. Then $f^{\alpha(1)}\wedge\cdots f^{\alpha(P(m))}$ appears in $\bigwedge^{P(m)}(S/I_{X})_{m}^{*}$ with a nonzero Plücker co-ordinate if and only if $x^{\alpha(1)},\dots,x^{\alpha(P(m))}$ form a basis of $(S/I_{X})_{m}$. Since $\rho$ is a co-character of the special linear group, the weights of all monomials of $S_{m}$ sum up to zero. Also, $\rho$ acts on $\mathbb{P}(\bigwedge S_{m})$ and $\mathbb{P}(\bigwedge S_{m}^{*})$ with opposite weights. Hence we have

[TABLE]

2. Decomposition of states

Let $G={\mathrm{G}L}(V)$ and let $V_{i}$, $i=1,\dots,\nu$, be vector subspaces of $V$ that span $V$. Note that $V=\sum_{i=1}^{\nu}$ is not necessarily a direct sum. Then

[TABLE]

For a notational convenience, we let $W$ denote $S^{m}V$, $W_{j}=S^{m}V_{j}$ for $1\leq j\leq\nu$ and

[TABLE]

Let $R$ be a maximal torus of ${\mathrm{G}L}(V)$ which preserves the subspaces $V_{i}$. Then one can choose a basis of $\mathcal{B}=\{v_{1},\dots,v_{M}\}$ of $V$ diagonalizing the $R$-action such that $V_{j}$ is the linear subspace spanned by $\{v_{M_{j}},v_{M_{j}+1},\dots,v_{M_{j}^{\prime}}\}$. We identify $GL(V_{j})$ with the subgroup of $GL(V)$ which preserves $V_{j}$ and acts trivially on $\mathrm{Span}\{v_{i}\,|i<M_{j},i>M_{j}^{\prime}\}$.

Let $\chi_{i}$ be the character of $R$ determined by $t.v_{i}=\chi_{i}(t)v_{i}$. Set $R_{j}=R\cap{\mathrm{G}L}(V_{j})$. Then there is a natural projection $\pi_{j}:R\to R_{j}$ defined by

[TABLE]

Then $\pi_{j}$’s induce injective group homomorphisms $\pi_{j}^{*}:X(R_{j})\hookrightarrow X(R)$. We shall identify $X(R_{j})$ with its image in $X(R)$ under $\pi_{j}^{*}$.

Proposition 2.1**.**

Let $I$ be a subspace of $W$, $I_{j}=I\cap W_{j}$, and suppose that the sum $I=\sum_{j=1}^{\nu+1}I_{j}$ is direct. Let $\dim I=N$ and $\dim I_{j}=N_{j}$, $1\leq j\leq\nu+1$. We have the decomposition of states

[TABLE]

Proof.

Let $\xi$ be an affine point over $\left[\bigwedge^{N}I\right]\in\mathbb{P}(\bigwedge^{N}S^{m}V)$, and let $\xi_{j}$ be an affine point over $\left[\bigwedge^{N_{j}}I_{j}\right]$ in $\bigwedge^{N_{j}}W_{j}$. The affine point $\xi_{j}$ generates the one-dimensional subspace $\bigwedge^{N_{j}}I_{j}$. Let $j\leq\nu$. Consider the $R$-weight decomposition of $\xi_{j}$. Since $I_{j}$ is contained in the $R$-module $W_{j}=S^{m}V_{j}$, the $R$-weight decomposition of $\xi_{j}$ is precisely the $R_{j}$-weight decomposition i.e.

[TABLE]

Since the sum $I=\sum_{j=1}^{\nu+1}I_{j}$ is direct, we have

[TABLE]

and hence the $R$-weight decomposition of $\xi$ is given as

[TABLE]

A summand $\bigotimes_{j=1}^{\nu+1}(\xi_{j})_{\chi^{(j)}}$ has a weight $\sum_{j=1}^{\nu+1}\chi^{(j)}\in X(R)$ and it is a state of $\xi$ if and only if the weight vector $\sum_{j=1}^{\nu+1}(\xi_{j})_{\chi^{(j)}}$ is nonzero if and only if $\chi^{(j)}$ is a state of $\xi_{j}$, $\forall j$. It follows that every state of $\xi$ is a sum of states of $\xi_{i}$’s and vice versa. ∎

2.1. Proof of Theorem 1.2

We shall now deduce Theorem 1.2 from Proposition 2.1. Let $X=X_{1}\cup X_{2}\subset\mathbb{P}(V^{*})$ be a chain of subvarieties $X_{i}$ and suppose that there exists a homogeneous coordinate system $x_{0},\dots,x_{\ell},\dots,x_{n}\in V$ such that

[TABLE]

We also assume that $X_{1}\cap X_{2}\neq\emptyset$, and that $X,X_{1},X_{2}$ are cut out by saturated homogeneous ideals $I_{X_{i}}$ and $I_{X}=I_{X_{1}}\cap I_{X_{2}}$.

Let $V_{1}$ (resp. $V_{2}$) be the subspace of $V$ spanned by $\{x_{0},\dots,x_{\ell}\}$ (resp. $\{x_{\ell},\dots,x_{n}\}$). Let $W=S^{m}V$, $W_{i}=S^{m}V_{i}$ for $i=1,2$, and $W_{3}=\sum_{i+j=m,ij\neq 0}S^{i}V^{\prime}_{1}\otimes S^{j}V^{\prime}_{2}$ where $V^{\prime}_{1}=\sum_{i=0}^{\ell-1}kx_{i}$ and $V^{\prime}_{2}=\sum_{i=\ell+1}^{n}kx_{i}$. Evidently we have $W=\sum_{i=1}^{3}W_{i}$. Let $R$ be the maximal torus of ${\mathrm{G}L}(V)$ diagonalized by $x_{0},\dots,x_{n}$ and $R_{i}=R\cap{\mathrm{G}L}(V_{i})$ for each $i$, where ${\mathrm{G}L}(V_{i})$ is identified with a suitable subgroup of ${\mathrm{G}L}(V)$ as in the discussion preceding Proposition 2.1. Of course, $R_{1}$ (resp. $R_{2}$) is identified with the maximal torus of ${\mathrm{G}L}(V_{1})$ (resp. ${\mathrm{G}L}(V_{2})$) diagonalized by $x_{0},\dots,x_{\ell}$ (resp. $x_{\ell},\dots,x_{n}$).

For each $m\geq 2$, we have

[TABLE]

We claim that the property of coordinates ($\dagger$) implies that this is a direct sum decomposition. Indeed, since $W_{1}\cap W_{2}$ is the $1$-dimensional space $k.x_{\ell}^{m}$ and $x_{\ell}^{m}$ does not vanish at $\{p\}=X_{1}\cap X_{2}$, $I_{X}\cap W_{1}\cap W_{2}=0$. Since $W_{1}$ and $W_{2}$ meet $W_{3}$ trivially, the claim follows and we may apply Proposition 2.1.

Note that the three terms on the right hand side of (2) are $I_{X_{1}}\cap k[x_{0},\dots,x_{\ell}]_{m}$, $I_{X_{2}}\cap k[x_{\ell},\dots,x_{n}]_{m}$ and $T_{m}$ respectively, where $T=\langle x_{0},\dots,x_{\ell-1}\rangle\langle x_{\ell+1},\dots,x_{n}\rangle$. Then by Proposition 2.1, we have

[TABLE]

where $N$ and $N_{i}$ denote the appropriate dimensions. As observed in the introduction, we may naturally identify the characters $\prod_{i=0}^{n}\chi_{i}^{a_{i}}$ with the Laurent monomials $\prod_{i=0}^{n}x_{i}^{a_{i}}\in k(x_{0},\dots,x_{n})$, where $\chi_{i}$ is the projection $t=(t_{0},\dots,t_{N})\mapsto t_{i}$. Identifying the characters with monomials and taking the convex hull of both sides, we obtain Theorem 1.2 for the case $\ell=2$ from which, as observed in [HK], the general case follows by a simple induction.

Remark 2.2**.**

Note that since $T$ is a monomial ideal, $\Xi_{\left[\bigwedge^{N_{3}}T_{m}\right]}(R)$ consists of one point $\chi^{\tau}$ where $\tau=\sum_{\begin{subarray}{c}x^{\alpha}\in T_{m}\end{subarray}}\alpha$.

2.2. Decomposition of vertices

Let $P_{1},\dots,P_{r}$ be polytopes in $\mathbb{R}^{n}$. In general, every face $F$ of the Minkowski sum $\sum_{i=1}^{r}P_{i}$ has a unique decomposition $F=\sum_{i=1}^{r}F_{i}$ into a sum of faces $F_{i}$ of $P_{i}$. The converse is easily seen to be false: If the origin ${\bf 0}$ is a vertex of a polytope $P$, then ${\bf 0}+v=v$ is not a vertex of $2P$ for any nonzero vertex $v$ of $P$. The following lemma guarantees that vertices always sum up to be a vertex provided that the polytopes are positioned well enough.

Lemma 2.3**.**

Let $P_{1},P_{2}$ be polytopes in $\mathbb{R}^{n}$. Suppose that $P_{1}$ and $P_{2}$ are contained in affine hyperplanes $H_{1}$ and $H_{2}$ respectively such that $H_{1}\cap H_{2}$ is of dimension one. Then the vertices of the Minkowski sum $P_{1}+P_{2}$ are precisely the sums of vertices of $P_{1}$ and $P_{2}$.

Proof.

It is evident that a vertex of $P_{1}+P_{2}$ is a sum of vertices of $P_{1}$ and $P_{2}$ since for any subsets $S_{1}$ and $S_{2}$ of $\mathbb{R}^{n}$, the sum of their convex hulls is the convex hull of their sum $S_{1}+S_{2}$. To prove the converse, we start by choosing affine coordinates $x_{1},\dots,x_{n}$ judiciously so that

[TABLE]

and

[TABLE]

Let $\{v_{1},\dots,v_{r}\}$ and $\{w_{1},\dots,w_{s}\}$ be the sets of vertices of $P_{1}$ and $P_{2}$, respectively. We aim to show that $v_{i}+w_{j}$ is a vertex of $P_{1}+P_{2}$, for any $i,j$. Suppose it is not the case - suppose without losing generality $v_{1}+w_{1}$ is not a vertex. Then there exist $\lambda_{ij}$ for $i=1,\dots,r,j=1,\dots,s$ such that $\sum_{i,j}\lambda_{ij}=1$, $0\leq\lambda_{ij}\leq 1$, $\lambda_{11}=0$ and

[TABLE]

By rearranging the terms, we have

[TABLE]

which implies that $x_{n_{1}}$ is the only nonzero coordinate of each side. Moreover, $1-\sum_{j}\lambda_{1j}\neq 0$ or $\sum_{i}\lambda_{i1}-1\neq 0$ since $\sum_{i,j}\lambda_{ij}=1$. Suppose the $1-\sum_{j}\lambda_{1j}\neq 0$ (the other case is proved similarly) and let $v_{i}=(v_{i0},v_{i1},\dots,v_{in_{1}})$. Then we have

[TABLE]

where $\mu_{ij}=\lambda_{ij}/(1-\sum_{j^{\prime}=1}^{s}\lambda_{1j^{\prime}})$ and $\sum_{\begin{subarray}{c}i\neq 1\\ 1\leq j\leq s\end{subarray}}\mu_{ij}=1$.

The $n_{1}$th coordinate also satisfies the above condition because

[TABLE]

which means that $v_{1}=\sum_{i\neq 1}\left(\sum_{j=1}^{s}\mu_{ij}\right)v_{i}$. But this is a contradiction since $v_{1}$ is a vertex of $P_{1}$. ∎

As an immediate corollary, we obtain:

Corollary 2.4**.**

Retain notations from Theorem 1.2. Let $\mathcal{V}_{i}$ denote the set of vertices of ${\mathcal{P}}_{m}(I_{X_{i}}\cap k[x_{n_{i-1}},\dots,x_{n_{i}}])$, $i=1,\dots,\ell$. Then the vertices of ${\mathcal{P}}_{m}(I_{X})$ are precisely

[TABLE]

Proof.

The $\ell=2$ case follows from Lemma 2.3: The state polytopes ${\mathcal{P}}_{m}(I_{X_{1}}\cap k[x_{0},x_{1},\dots,x_{n_{1}}])$ and ${\mathcal{P}}_{m}(I_{X_{2}}\cap k[x_{n_{1}},x_{n_{1}+1}\dots,x_{n}])$ are in the affine hyperplanes

[TABLE]

and

[TABLE]

respectively, where $Q_{1}(m)=\dim I_{X_{1}}\cap k[x_{0},x_{1},\dots,x_{n_{1}}]_{m}$ and $Q_{2}(m)=\dim I_{X_{2}}\cap k[x_{n_{1}},x_{n_{1}+1}\dots,x_{n}]_{m}$. Since $H_{1}\cap H_{2}$ is one dimensional, Lemma 2.3 applies. The general, $\ell\geq 2$ case follows by an induction. ∎

3. Decomposition of Hilbert-Mumford index

Retain the notations from Section 2.1. To prove Proposition 1.3, as in [HK] we shall assume that $\ell=2$ as the general case follows by a simple induction. We let $Q(m)=\dim(I_{X})_{m}$ and $P(m)=\dim(S/I_{X})_{m}=\dim S_{m}-Q(m)$. Likewise, $Q_{i}(m)=\dim(I_{X_{i}})_{m}$ and $P_{i}(m)=\dim S^{m}V_{i}-Q_{i}(m)$. Let $\rho$ be a 1-PS of $R$ and let $\rho_{i}$ be the induced 1-PS of $R_{i}$, $i=1,2$. These are obtained by composing with the projections $R\to R_{i}$. Recall that, if the sum of the $\rho$-weights is zero, the Hilbert-Mumford index is given by

[TABLE]

where $\langle\,\,,\,\,\rangle$ denotes the natural paring of the character group and the 1-PS group i.e. $\chi\circ\rho(t)=t^{\langle\chi,\rho\rangle}$ for any $t\in\mathbb{G}_{m}(k)$.

For any $\chi\in\Xi_{[I_{X}]_{m}}(R)$, due to ($\dagger\dagger$ ‣ 2.1), we have $\chi=\chi_{1}+\chi_{2}+\tau$, $\chi_{i}=\iota_{i}\circ\chi$ where $\iota_{i}:R_{i}\to R$ is the inclusion. And $\tau$ is the character with which $\rho$ acts on $\bigwedge^{\textrm{max}}\left(I\cap\sum_{i+j=m,ij\neq 0}S^{i}V_{1}\otimes S^{j}V_{2}\right)=T_{m}$ where $T=\langle x_{0},\dots,x_{\ell-1}\rangle\langle x_{\ell+1},\dots,x_{n}\rangle$.

Hence we have

[TABLE]

Clearly, the minimum of $\langle\chi,\rho\rangle$ is achieved precisely when each $\chi_{i}$ pairs minimally with $\rho$. Let $\rho^{\prime}$ be the 1-ps of ${\mathrm{S}L}(V)$ associated to $\rho$ i.e. if $r_{i}$ are the weights of $\rho$, then $\rho^{\prime}$ is the 1-ps with weights $r_{i}-w$ where $w$ is the average of the weight $\frac{1}{\dim V}\sum_{i=1}^{n}r_{i}$. Conflating a 1-ps with its weight vector, we may write $\rho=\rho^{\prime}+(w,w,\dots,w)$.

The minimum of $\langle\chi,\rho\rangle$ is achieved by

[TABLE]

where we used the multiplicative multi-vector notation $\chi^{\alpha}=\prod\chi_{i}^{\alpha_{i}}$ as in the discussion preceding Theorem 1.1. Note that

[TABLE]

Similarly, let $\rho_{i}^{\prime}$ denote the 1-ps of ${\mathrm{S}L}(V_{i})$ associated to $\rho_{i}$, $i=1,2$, whose weights are shifted by the average weight $w_{i}=\frac{\sum_{x_{j}\in V_{i}}\mathrm{wt}_{\rho}(x_{j})}{\dim V_{i}}$. Clearly, $\chi_{i}$ pairs with $\rho_{i}$ minimally if and only if it pairs with $\rho_{i}^{\prime}$ minimally, and

[TABLE]

Hence we have

[TABLE]

Substitute $\dim S^{m}V-Q(m)=P(m)$ and $\dim S^{m}V_{i}-P_{i}(m)=Q_{i}(m)$. And subsequently, substitute $\sum_{x^{\alpha}\in S^{m}V}\mathrm{wt}_{\rho}(x^{\alpha})$ for $mw\dim S^{m}V$ and $\sum_{x^{\alpha}\in S^{m}V_{i}}\mathrm{wt}_{\rho}(x^{\alpha})$ for $mw_{i}\dim S^{m}V_{i}$. Then we get

[TABLE]

since $S^{m}V_{1}\coprod(S^{m}V_{2}\setminus\{x_{\ell}^{m}\})\coprod T_{m}=S^{m}V$. Recall from the closing moment of the introduction that $[I_{X}]_{m}$ and $[I_{X}]_{m}^{*}$ have the same Hilbert-Mumford indices, and that $\rho$ acts with opposite weights on $\mathbb{P}(\bigwedge S_{m}^{*})$ in which the dual Hilbert points live. That is, if $w^{*}$ is the average of the weights for the $\rho$ action on $\mathbb{P}(\bigwedge S_{m}^{*})$, then $w*=-w$ so that the signs of the terms $wmP(m)$, $w_{i}mP_{i}(m)$ are reversed. Hence ($\ddagger$ ‣ 3) is precisely the assertion of Proposition 1.3 for the case $\ell=1$.