# Kyle: The Polytope Algebra (Tuesday, June 14) ## Defining the Polytope Algebra :::{.remark} Goal: define the *polytope algebra* and explore its properties. It generalizes the polytope group from previous talks and many analogous results hold, but has extra structure that keeps track of lower-dimensional data. It has applications to many fields: discrete, tropical, and toric geometry which arise in the study of Chow, \(\K\dash\)theory, etc. ::: :::{.definition title="?"} Let $\mcp$ be the free abelian group of all polytopes (not just full-dimensional ones) in $\RR^d$ with $d$ fixed, and consider relations 1. $[P \union Q] + [P \intersect Q] = [P] + [Q]$ 2. $[P+t] = [P]$ for any translation $t$ Let $\Pi$ be the quotient of $\mcp$ by these relations. ::: :::{.remark} One can express interiors of polytopes in this algebra: ![](figures/2022-06-14_10-12-12.png) One can also be replaced by $[P] + [P \intersect H] = [P \intersect H^+] + [P \intersect H^+]$. ::: :::{.remark} In $\Pi$, the multiplication is given by the Minkowski sum $[P][Q] \da [P+Q]$ where $P+Q=\ts{p+q\st p\in P, q\in Q}$. One can use the scissors relation to prove distributivity. An example of the Minkowski sum: ![](figures/2022-06-14_10-17-41.png) One can realize the sum by arranging all edges of $P$ and $Q$ sorted by slope. An example equation: ![](figures/2022-06-14_10-19-58.png) ::: :::{.warnings} One has to be careful with representatives: > Todo: missing picture of overlapping vs disjoint polygons. ::: :::{.remark} Any endomorphism $\phi$ of $\RR^d$ which is affine induces a map $\hat\phi: \Pi\to \Pi$. ::: :::{.corollary title="?"} For every $\lambda \in \RR_{\geq 0}$, the dilation by $\lambda$ map $\lambda: \RR^d\to \RR^d$ induces $\Delta(\lambda): \Pi\to \Pi$. ::: ## Filtration and the Rational Structure :::{.remark} $\Pi$ is almost a graded commutative algebra! Note that $\Pi_0$ is defined as sums of points in $\RR^d$, and $\Pi_0 \cong \ZZ$. ::: :::{.proposition title="?"} Let $Z_1$ be the ideal generated by $[P] - 1$ for $P$ nonempty. $\Pi$ has a decomposition $\Pi = \Pi_0 \oplus Z_1$, and $Z_1$ is an ideal inside $\Pi$, witnessed as $\Pi_0 = \ker \Delta(0)$. ::: :::{.proof title="?"} In McMullen, - Proved for simplices, i.e. $\Delta(0)([\mathrm{simplex} ] - 1) = 0$. - Extend to polytopes since every polytope is triangulable. They then conclude since $\Delta(0)$ is equal on 0-dimensional simplices. ::: :::{.lemma title="?"} For $r>d$, \[ ([P] - 1)^r = 0 .\] ::: :::{.example title="?"} Let $d=2, r= 3$ and let $P$ be a simplex. What is $[P]^3$? ![](figures/2022-06-14_10-39-46.png) By standard arguments in Ehrhart theory (??), one can show the relation above: expand $([P] - 1)^3 = [P]^3 -3[P]^2 + 3[P] - 1$, so look at inclusion-exclusion on this diagram: ![](figures/2022-06-14_11-17-21.png) > Todo: wrong diagram! ::: ## Rational Structures :::{.remark} We'll show $Z_1$ is torsionfree and divisible. ::: :::{.lemma title="?"} If $x\in Z_r$ then $\Delta(n)x - n^rx\in Z_{r+1}$ for any positive integer $n$. ::: :::{.proof title="?"} Suppose $x= ([P] - 1)^r$, then note \[ \Delta(n)([P] - 1) = \sum_{1\leq k\leq d} {n\choose k} ([P] - 1)^k .\] Taking $r$th powers on both sides yields \[ \qty{ \Delta(n)([P] - 1) }^r \Delta(n)\qty{ ([P] - 1)^r } = \Delta(n)x .\] Note that $([P] - 1)^i \in Z_{r+1}$ for large $i$, and ${n \choose 1}([P] - 1)^r = n^r x$. ::: :::{.lemma title="?"} $Z_1$ is torsionfree. ::: :::{.proof title="?"} Let $x\in Z_1$ with $nx=0$, it STS $x=0$. Check $\Delta(n)x = \delta(n) - n^{r-1} n x\in Z_{r+1}$, noting $nx=0$ by assumption. This shows $\Delta(n)x\in Z_{r+1}$. For all $y\in Z_j$, we have $\Delta(\lambda)y\in Z_j$ for any $\lambda$. Thus $x= \Delta(n\inv)\delta(n)x$, so $x$ is in every filtration level and thus zero. ::: :::{.lemma title="?"} $Z_1$ is divisible. ::: :::{.proof title="?"} Let $x\in Z_1$ and $m\geq 2$ be an integer, and proceed inductively. The base case is $x\in Z_d$; write \[ x = \Delta(m) \delta(m\inv) x = mm^{d-1} \delta(m\inv) x ,\] so $m\inv x = m^{d-1} \Delta(m\inv)x$. Now induct: let $y = mm^{r} \delta(m\inv) x\in Z_{r+1}$, so $m\inv x = m^{r-1} \Delta(m\inv) x + m\inv y\in Z_r$. ::: :::{.theorem title="?"} $Z_1$ has a rational structure, i.e. an intrinsic $\QQ\dash$algebra structure. ::: :::{.remark} $\Pi$ is the simplest $\lambda$ ring! :::