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For a representation on which a commuting torus or Cartan action is semisimple, the weight decomposition is the direct sum of simultaneous weight spaces. A finite-dimensional representation of a compact torus can be made unitary by Haar measure averaging; its commuting unitary matrices are simultaneously diagonalizable. Each diagonal character is an integral weight of a representation. The decomposition makes tensor and exterior-power weights additive. For nonsemisimple actions, simultaneous generalized eigenspaces give a generalized weight decomposition instead.

A global ordered framing selects the displayed $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$-sheeted cover inside the unordered Stiefel bundle. It is a trivial cover because the frame labels are global. Selecting subsets of a direct-sum framing decomposes the cover as the disjoint union of fiber products of the two summands' subset covers.

In a local parallel orthonormal frame a holomorphic section has holomorphic coefficients $f_j$. Its squared norm satisfies ${\partial }_z{\partial }_{\bar{z}}{\sum }_j|f_j{|}^2={\sum }_j|f_j^{\prime }{|}^2\ge 0$. On a compact connected complex curve, integration of the Laplacian makes the right side zero. Thus every coefficient is locally constant and the section is parallel. A global parallel section is determined by a vector fixed by all monodromy operators, and every such vector gives a global section.

The group of unitary bundle gauge transformations under composition. It acts on connections and their solution spaces. The stabilizer of a connection is the centralizer of its holonomy acting by parallel automorphisms, so a quotient can have singularities when this stabilizer increases.

// Destination: geometry-and-topology.bigb

Set the lower half-space load to $q_b={\mu }_b{p}_{\beta ,b}$ using the causal vertical wavenumber branch, which excludes a wave arriving from depth infinity. Apply the uniform-layer SH impedance update upwards through every layer, keeping particle velocity and traction continuous. For upper half-space impedance $q_a$, incident velocity $V_0$ and reflected velocity $R{V}_0$, the top values are $V=V_0(1+R)$ and $T=-q_a{V}_0(1-R)$. Their ratio gives the displayed reflection coefficient, incorporating every underlying interface and multiple reflection.

A Slater determinant extends two-particle antisymmetrization to $N$ fermions. Here $x_i$ contains position and spin or other internal variables. Distinct orthonormal one-particle states give the normalized wavefunction displayed in the title. Repeating a one-particle state repeats a column of the determinant, so the wavefunction vanishes. Exchanging particle slots exchanges rows and reverses its sign. A general fermionic quantum state can be a quantum superposition of Slater determinants; it need not be one Slater determinant.

At the centre of geodesic normal coordinates, expand the square of the Twisted Dirac operator. The diagonal terms give the rough Laplacian; the off-diagonal terms are ${\sum }_{i<j}c(e^i)c(e^j)[{\nabla }_i,{\nabla }_j]$. The tensor-product vector-bundle curvature is the sum of spin curvature and $F^W$. Substituting spin curvature $\frac{1}{4}{\sum }_{a,b}⟨R(e_i,e_j)e_a,e_b⟩c(e^a)c(e^b)$ and using the Clifford algebra relations yields the scalar term $\mathrm{Scal}/4$: the four-generator contribution vanishes by the algebraic Bianchi identity and the contracted terms are the scalar curvature. This proves the displayed identity at every point.

For the defining representation of the special linear Lie algebra ${\mathfrak{sl}}_3$, the crystal basis is $1\stackrel{1}{\to }2\stackrel{2}{\to }3$, with weights ${\epsilon }_i$ restricted to trace-zero diagonal matrices. Its dual is $\bar{3}\stackrel{2}{\to }\bar{2}\stackrel{1}{\to }\bar{1}$, with weights $-{\epsilon }_i$. Using the crystal tensor-product rule with lowering on the first factor when ${\phi }_i(a)>{\epsilon }_i(b)$, the tensor square has highest vertices $1\otimes 1$ and $1\otimes 2$, of highest weights $2{\omega }_1,{\omega }_2$, generating components of six and three vertices. The defining tensor dual has highest vertices $1\otimes \bar{3}$ and $1\otimes \bar{1}$, generating components of eight and one vertices and weights ${\omega }_1+{\omega }_2,0$. These encode the symmetric square plus exterior square, and the trace-free endomorphisms plus scalar identity, respectively. A crystal vertex labelled by a tensor pair need not represent that same undeformed ordinary simple tensor; the actual invariant vector is ${\sum }_i{e}_i\otimes e_i^{\ast }$.

For a compact string embedding map coordinate, T-duality preserves its left-moving part and reverses its right-moving part. Consequently ${\partial }_{\tau }\tilde{X}={\partial }_{\sigma }X$ and ${\partial }_{\sigma }\tilde{X}={\partial }_{\tau }X$. A Neumann boundary condition becomes a Dirichlet boundary condition and conversely. A D-brane wrapping the circle loses one spatial worldvolume dimension; a D-brane transverse to it gains one in the dual description.

For $d(p,q)=(p+1)(q+1)(p+q+2)/2$, the exact largest constituent dimension of a mixed SU(3) tensor representation isEvery constituent highest weight lies below $k{\omega }_1+l{\omega }_2$, so its labels have the form $(p,q)=(k-2a+b,l+a-2b)$ for nonnegative integers $a,b$. If both are positive, reducing each by their minimum increases both $p,q$ and strictly increases the dimension. The resulting boundary labels occur: antisymmetrize disjoint upper pairs into lower indices, or lower pairs into upper indices, then take the symmetric traceless component. This proves the maximum formula. For $(k,l)=(5,0)$, the fully symmetric dimension is 21 but $(3,1)$ has dimension 24. For $k=l$, the fully symmetric traceless dimension $(k+1{)}^3$ is the maximum.

The degree-three monomials $z_1^a{z}_2^b{z}_3^c$, with $a+b+c=3$, form a basis of this ten-dimensional highest-weight representation. Its highest monomial is $z_1^3$, and $E_{21}=z_2{\partial }_{z_1}$ and $E_{32}=z_3{\partial }_{z_2}$ generate every basis monomial. Its Dynkin labels are $(3,0)$, and the weight of a monomial is $(a-b,b-c)$. In coordinates $(h_1,(h_1+2h_2)/\sqrt{3})$, the ten weights form four horizontal rows of lengths $4,3,2,1$, all with weight multiplicity one. The baryon decuplet realizes this representation of approximate flavor symmetry.

// Target: special-relativity.bigb

For $n\ge 2$, contraction with the invariant symmetric bilinear form splits off the trivial line in the symmetric square of the defining special orthogonal group representation. Its null space is the traceless symmetric square. It contains a highest-weight vector of weight $2e_1$, and the Weyl dimension formula gives $(n+1)(2n-1)$, exactly its dimension. Complete reducibility of compact-group representations therefore makes that null space irreducible. The entire symmetric square is reducible because of the invariant line.

For the defining $n$-dimensional special linear Lie algebra module, with $n\ge 2$, write $T_i(p)=e_i\otimes p(y_1,\dots ,y_n)$. The equivariant contraction is $C(T_i(p))={\partial }_{y_i}p$. The map $\iota (f)={\sum }_i{e}_i\otimes y_{i}f$ is also equivariant, since ${\sum }_i{e}_i\otimes y_i$ is an invariant identity tensor. Direct differentiation gives $C\iota =(n+1)I$, so the displayed direct sum follows. Its trace-free part is an irreducible representation of highest weight ${\omega }_1+2{\omega }_{n-1}$. Its highest-weight vector is $e_1\otimes y_n^2$; the Weyl dimension formula gives dimension $n(n-1)(n+2)/2$, exactly $\dim \ker C$.

// Target: semisimple-lie-algebra.bigb

In a nontrivial symmetric balanced incomplete block design, $\det (N{)}^2=r^2(r-\lambda {)}^{t-1}$. If $t$ is even, $t-1$ is odd. For every prime, the exponent in the left side is even and that in $r^2$ is even, so its exponent in $r-\lambda$ must be even. Thus $r-\lambda$ is a perfect square. This supplies an elementary necessary existence condition without claiming it is sufficient.

For an axisymmetric random velocity dispersion tensor with ${\Pi }_{11}={\Pi }_{22}=M{\sigma }_0^2$ and ${\Pi }_{33}=M{\sigma }_0^2(1-\delta )$, contracting with the line-of-sight unit vector $(\sin i,0,\cos i)$ gives the displayed mass-weighted velocity dispersion. A major-axis rotation amplitude projects as $V_r=V\sin i$. By contrast, an azimuthally averaged line-of-sight streaming root-mean-square is $V\sin i/\sqrt{2}$ when $V^2$ is the full in-plane streaming mean square. Aperture weights and the definition of the measured rotation statistic must match the intrinsic statistic.

In normalization $Q=T_3+Y$, one Standard Model generation has left quark and lepton doublet hypercharges $1/6$ and $-1/2$, and right singlet hypercharges $2/3,-1/3,-1$. Including colour multiplicity, ${\mathrm{tr}}_{L}Y=6/6-2/2=0$. On every left doublet $T_3^2=I/4$, and all right fields have $T_3=0$, so the mixed weak-hypercharge gauge anomaly vanishes. The cubic left trace is $6(1/6{)}^3+2(-1/2{)}^3=-2/9$, and the right trace is $3(2/3{)}^3+3(-1/3{)}^3+(-1{)}^3=-2/9$. Their difference is zero, proving generation-by-generation cancellation of these two anomalies.

Start the symmetric stochastic attrition process at $(N,\lfloor \alpha N\rfloor )$, with $0<\alpha <1$, and let $X^N$ be the final first-population size divided by $N$. The bound holds for all sufficiently large $N$, depending on $\alpha ,\delta >0$. At absorption, one coordinate is zero. If the first survives, the terminal squared difference is $N^2(X^N{)}^2$; if it is extinguished, that difference is nonpositive. In either case the indicated deviation forces an error of at least $\sqrt{1-{\alpha }^2}\delta N^2$ from the limiting squared invariant. The initial rounding error and total conditional drift are only $O(N)$. Apply Azuma-Hoeffding inequality to the centered squared-population near-invariant for stochastic attrition, using its $2N$ increments bounded by $4N$, to obtain the displayed exponent.

A spinor representation is a representation of a Spin group, or infinitesimally of a Special orthogonal Lie algebra, constructed from a Clifford algebra. In even dimensions it can split into two chiral spinor representations.

A fixed volume form makes wedge pairing on ${\Lambda }^2{\mathbb{R}}^4$ a symmetric form of signature $(3,3)$. The exterior-square action of special linear group matrices preserves it. Its kernel is $\{\pm I\}$: a transformation acting trivially on every bivector preserves every two-plane and hence every line, so is scalar, with scalar square one. The derivative is injective and both groups have dimension fifteen, so the connected image is the full identity component. This is a concrete real spin-group cover.

Congruence $X\mapsto AX{A}^T$ on real symmetric two-by-two matrices preserves the quadratic form $-\det X$ of signature $(2,1)$ when $\det A=1$. Its kernel is $\{\pm I\}$ and its derivative is injective: $aX+X{a}^T=0$ for every symmetric $X$ implies $a=0$. Both Lie groups have dimension three, so the connected image is the full identity component. Also $A^{T}JA=(\det A)J$ identifies $Sp(2,\mathbb{R})$ with $SL(2,\mathbb{R})$.