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The capacity of a finite set is the total mass of its equilibrium measure of a finite set:Capacity is monotone under inclusion, though strict inclusion need not give strict inequality.

The Coxeter complex has faces represented by cosets $w{W}_J$ of standard parabolic subgroups, ordered by reverse inclusion. Geometrically, its maximal simplices are the chambers of the Coxeter arrangement intersected with a sphere.

For the closed fundamental chamber $D\subseteq V^{\ast }$, the Tits cone isIt equals all of $V^{\ast }$ exactly when the finite-rank Coxeter group $W$ is finite.

A connected Eilenberg–MacLane space $K(G,n)$, $n\ge 1$, has ${\pi }_n\cong G$ and all other positive-degree homotopy groups trivial. The group $G$ must be abelian when $n\ge 2$, but may be nonabelian when $n=1$. For an abelian group, the Dold–Kan correspondence constructs a simplicial model from the chain complex concentrated in degree $n$. A classifying space of a discrete group gives the general $K(G,1)$ case.

The Hodge number of a compact complex manifold is $h^{p,q}={\dim }_{\mathbb{C}}{H}_{\bar{\partial }}^{p,q}(X)$.

For a compact Kähler manifold, these dimensions are the graded dimensions in its Hodge structure.

The tensor product crystal uses the $i$-string data ${\epsilon }_i$ and ${\phi }_i$ to decide on which tensor factor a Kashiwara operator acts. Its connected components are highest-weight crystals and encode the decomposition of the tensor product representation.

This tensor-product operation combines one crystal basis with another, with factor selection governed by their string parameters.

On a crystal, ${\tilde{f}}_i$ follows an edge of color $i$ and ${\tilde{e}}_i$ follows that edge backwards.

These maps are the raising and lowering operations of a crystal basis.

The type $E(p,q,k)$ graph is a simply-laced tree with one trivalent vertex and arms containing $p$, $q$, and $k$ vertices beyond that vertex.

An irreducible Coxeter system is one whose Coxeter graph is connected. The connected components of a Coxeter graph give the direct-product decomposition of its Coxeter group.

The Bockstein homomorphism is the connecting homomorphism associated with a short exact sequence of coefficient groups. For $0\to \mathbb{Z}/n\to \mathbb{Z}/n^2\to \mathbb{Z}/n\to 0$, it has degree $-1$ on homology and degree $+1$ on cohomology.

A closed subscheme of $X$ is a scheme $Z$ together with a closed immersion $Z\hookrightarrow X$, usually identified with its image and its quotient structure sheaf.

The Ricci form is the real curvature form $i{F}_{\nabla }$ of the Chern connection on $T^{1,0}X$. In complex dimension one, $\rho =K\omega =\frac{1}{2}\mathrm{Scal}\omega$.

For the hyperplane class $H$,and $A_k({\mathbb{P}}^r)\cong \mathbb{Z}$ is generated by a linear ${\mathbb{P}}^k$.

A Čech coboundary is the image of a cochain in the preceding degree. Multiplicatively, a zero-cochain $(h_i)$ changes a one-cocycle by $g_{ij}\mapsto h_i^{-1}{g}_{ij}{h}_j$.

A Čech cochain is a cocycle when its alternating coboundary vanishes. For a multiplicative one-cochain $(g_{ij})$, this says $g_{ij}{g}_{jk}=g_{ik}$ on triple intersections.

A Fourier-space variance $P_k$ in three dimensions is scale invariant when the dimensionless power per logarithmic wavenumber interval,is independent of $k$. For a massless inflaton mode at late times,so ${\Delta }^2(k)$ is constant.

Extending an algebraic-concordance class from $\mathbb{Q}$ to a p-adic field ${\mathbb{Q}}_p$ detects torsion invisible over the real numbers. For $p\equiv 3(\mathrm{mod}4)$, an odd-dimensional second-residue form generates the order-four part of the local Witt group.

An isometric structure consists of a finite-dimensional vector space $V$, a nonsingular symmetric bilinear form $Q$, and a $Q$-isometry $T$ with the required nondegeneracy at $\pm 1$. Metabolic isometric structures are quotiented out to form ${\mathcal{W}}_F$.

The knot determinant is $|{\Delta }_K(-1)|=|\det (A+A^T)|$. It is also the order of the first homology of the two-fold cover of $S^3$ branched over $K$.

The affine plane over a field $k$ is the affine scheme $\mathrm{Spec}k[x,y]$.

The two-dimensional scheme has an affine space of rational points over its base field; its scheme structure also records non-rational points and local rings.