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For $V={\mathbb{C}}^{2n}$, $n\ge 3$, the symmetric square splits into an invariant line and its irreducible traceless part of highest weight $2{\epsilon }_1$. The exterior square of the defining orthogonal representation is the adjoint module of highest weight ${\epsilon }_1+{\epsilon }_2$. For $n=2$, the exterior square splits instead into highest weights ${\epsilon }_1-{\epsilon }_2$ and ${\epsilon }_1+{\epsilon }_2$. This exception reflects that so4 Lie algebra is semisimple but not simple.

A microchannel plate detector uses many narrow biased channels to amplify an incoming detection event. An incident particle or released electron can initiate secondary-electron emission; repeated wall impacts multiply the electrons into a measurable output pulse.

Position-sensitive readout records where the amplified charge arrives, and timing electronics records its arrival time. In the helium atom-correlation experiment, a plate combined with a delay-line anode registered individual arrivals after expansion of a metastable atomic cloud.^[ref] Neutral-atom detection relies on the prepared internal state and its surface interaction, not only on the atom's lack of charge. Efficiency, spatial resolution and multiple-hit response enter the inferred pair correlations; an amplification pulse is not a direct image of an atomic trajectory.

In the pp-I limit, helium-3 relaxation time is ${\tau }^{-1}=n_1\sqrt{2{\lambda }_{11}{\lambda }_{33}}$. At fixed proton number density, each thermonuclear reaction rate coefficient obeys $\lambda (T)/\lambda (T_0)=(T/T_0{)}^{-2/3}\exp [-\eta (T_0)((T_0/T{)}^{1/3}-1)]$. Taking their geometric mean gives the displayed relaxation ratio with $\bar{\eta }=[{\eta }_{11}(T_0)+{\eta }_{33}(T_0)]/2$. A local power-law approximation instead gives $\tau \propto T^{-(\alpha +\beta )/2}$. These are estimates rather than a complete radial model: the proton number density, composition and pp-II branch change along a real stellar composition profile.

Under T-duality, the positions $y_i$ of D-branes transverse to a circle of radius $R$ become diagonal Wilson lines on branes wrapping the dual circle $R^{\prime }={\alpha }^{\prime }/R$. Their holonomies are $\exp (i2\pi R^{\prime }{A}_i)=\exp (i{y}_i/R)$. An Open string stretched between D-branes with displacement $y_j-y_i+2\pi Rw$ becomes a string with shifted compact momentum $w/R^{\prime }+A_j-A_i$. The stretching contribution and the shifted-momentum contribution to its squared mass are identical.

Take two adjacent columns of a numbered Young tableau, with a bottom segment $A$ of the left column and a top segment $B$ of the right column such that $|A|+|B|$ exceeds the height of the left column. Alternating over permutations of $A\cup B$, modulo permutations preserving $A$ and $B$ separately, gives a signed relation among polytabloids. The full alternation vanishes because each tabloid has two of these entries in one row. Solving for a nonstandard polytabloid replaces its offending row inversion by earlier column-standard tableaux. The relation is proved over the free integral tabloid lattice, where cancellation and division by nonzero integer factors are valid; reduction then makes it valid in every characteristic. Together with column antisymmetry these relations give the standard polytabloid basis.

For a monad $(T,\eta ,\mu )$, the map ${\eta }_{TA}:TA\to TTA$ is a split equalizer of ${\eta }_{TTA},T{\eta }_{TA}$ with retraction ${\mu }_A$ and splitting $T{\mu }_A$. The map $T{\eta }_A:TA\to TTA$ is a split equalizer of $TT{\eta }_A,T{\eta }_{TA}$ with retraction ${\mu }_A$ and splitting ${\mu }_{TA}$. Each assertion follows from the two unit laws and naturality of the unit or multiplication. No assumption that $T$ preserves arbitrary equalizers is needed.

Every self-adjoint operator on a separable complex Hilbert space is diagonal modulo a compact operator. For a bounded operator the compact self-adjoint error can have arbitrarily small Hilbert-Schmidt norm. In a cyclic spectral representation, partition a bounded real interval into $N$ intervals of width at most $\delta$ and successively bisect. Orthogonal step-function differences give basis vectors supported in the parent intervals. A diagonal value chosen in each support interval gives total squared column error at most $3N{\delta }^2$. Since $N=O({\delta }^{-1})$, this tends to zero. Apply the construction to countably many cyclic summands with summable squared error bounds. For an unbounded operator first split into bounded spectral intervals, obtaining a bounded compact error on the same domain.

For induction of $S^{\mu }⊠S^{\lambda }$ to the symmetric group on $|\mu |+|\lambda |$ letters, the Specht filtration multiplicity of $S^{\nu }$ is the number of Semistandard Young tableaux of skew shape $\nu /\mu$, content $\lambda$, and lattice word read from right to left in each row, taking rows from top to bottom. In characteristic zero these are irreducible constituents; in positive characteristic they remain Specht factors rather than simple constituents. Content $(1)$ gives the add-one-cell induction rule.

A positive-rank holomorphic vector bundle on a compact Riemann surface is semistable when every nonzero proper holomorphic subbundle has slope at most its own. Equivalently one tests coherent subsheaves: saturation on a smooth curve preserves rank, increases degree, and gives a subbundle with torsion-free quotient. Strict inequality defines stability, which is stronger.

Fix a Hermitian metric on a holomorphic vector bundle $E$ over a compact Riemann surface and a positive area form $\omega$. Write $i{F}_E=K\omega$ for its Chern curvature. If every eigenvalue of the Hermitian endomorphism $K$ is at most $C$, the curvature formula for a holomorphic subbundle gives $\mathrm{deg}F\le \mathrm{rank}(F)C{\int }_X\omega /(2\pi )$. The second-fundamental-form term is nonpositive. Compactness supplies a finite $C$ independent of the subbundle. If $F_E=0$, every subbundle has nonpositive degree.

Let $t_{ij}=(e_i\otimes e_j+e_j\otimes e_i)/2$, and let $f_i$ be the dual basis. The equivariant contraction sends $t_{ij}\otimes f_k$ to ${\delta }_{ik}{e}_j+{\delta }_{jk}{e}_i$. Its kernel has a weight basis consisting of $t_{ii}\otimes f_k$ for $i\ne k$, $t_{ij}\otimes f_k$ for three distinct indices with $i<j$, and $t_{ij}\otimes f_j-\frac{1}{2}{t}_{ii}\otimes f_i$ for $i\ne j$. The three families have sizes six, three and six. This is the irreducible fifteen-dimensional highest-weight representation with highest weight $2L_1-L_3$.

For a smooth algebraic variety of dimension $r$ over a field, the sheaf of Kähler differentials over a field is locally free of rank $r$ and defines its algebraic cotangent bundle. Its dual is the tangent sheaf of a scheme. For a smooth hypersurface of degree $d$ in ${\mathbb{P}}^n$, the Conormal exact sequence for Kähler differentials is the short exact sequence $0\to {\mathcal{O}}_X(-d)\to {\Omega }_{{\mathbb{P}}^n}^1{|}_X\to {\Omega }_X^1\to 0$.

At the self-dual circle, left and right dimension-one currents generate $SU(2{)}_L\times SU(2{)}_R$. The compact-coordinate vertex $\partial Y_L\bar{\partial }{Y}_R$ is a Cartan–Cartan component of a scalar field in the $(3,3)$ representation. Its expectation value changes the circle radius and leaves $U(1{)}_L\times U(1{)}_R$ unbroken. The four extra charged vectors acquire the displayed mass, and four scalar field polarizations become their longitudinal modes. This realizes the Higgs mechanism geometrically. In a bosonic string the ever-present tachyon and additional scalar field instabilities prevent interpreting the whole family as a stable vacuum.

At $R=\sqrt{{\alpha }^{\prime }}$, a chiral compact boson has operator product expansion $Y_L(z)Y_L(w)\sim -{\alpha }^{\prime }\log (z-w)/2$. The currents $J^3=i\partial Y_L/\sqrt{{\alpha }^{\prime }}$ and $J^{\pm }=:e^{\pm 2i{Y}_L/\sqrt{{\alpha }^{\prime }}}:$ have conformal weight one. Contractions give $J^3(z)J^3(w)\sim 1/[2(z-w{)}^2]$, $J^3(z)J^{\pm }(w)\sim \pm J^{\pm }(w)/(z-w)$ and $J^{+}(z)J^{-}(w)\sim 1/(z-w{)}^2+2J^3(w)/(z-w)$. These are the level-one SU(2) Lie algebra currents. The right-moving copy supplies a second factor. Combining a current with a noncompact coordinate derivative gives a massless string vertex operator for each of the six vectors; four are additional to the generic two Cartan vectors.

For the unitary group $U(n)$ with $n\ge 3$, the third tensor power of its defining representation decomposes as displayed. The first two summands have highest weights $(3,0,\dots )$ and $(1,1,1,0,\dots )$ and dimensions $\left(\genfrac{}{}{0ex}{}{n+2}{3}\right)$ and $\left(\genfrac{}{}{0ex}{}{n}{3}\right)$. The mixed summand has highest weight $(2,1,0,\dots )$ and dimension $n(n^2-1)/3$. In the three-dimensional weight space spanned by permutations of $e_1\otimes e_1\otimes e_2$, the vectors killed by all positive root vectors have coefficient sum zero, giving multiplicity two. For $n=2$, the exterior cube vanishes and $V_{(2,1)}=\det \otimes {\mathbb{C}}^2$.

The Verma module $M(0)$ has basis $v_n=f^n{v}_0$, with $h{v}_n=-2n{v}_n$, $f{v}_n=v_{n+1}$ and $e{v}_n=n(1-n)v_{n-1}$. Its submodule spanned by $v_n$ for $n\ge 1$ is $M(-2)$, and the quotient is the trivial one-dimensional module. A complementary submodule would be trivial and hence lie in the zero-weight space $\mathbb{C}{v}_0$, but $f{v}_0=v_1\ne 0$. Therefore the displayed extension does not split. This explains why the finite-dimensional hypothesis in the Weyl complete reducibility theorem is essential.

For $N=q-1$, write a diagonal-torus character as $\theta (\mathrm{diag}(a,b,(ab{)}^{-1}))=\chi (a{)}^r\chi (b{)}^s$, with $\chi$ a generator of the character group of ${\mathbb{F}}_q^{\times }$. Its Weyl stabilizer is trivial exactly when $r,s,r-s$ are all nonzero modulo $N$ and, if $3\mid N$, the unordered pair $\{r,s\}$ is not $\{N/3,2N/3\}$. The first conditions exclude transpositions; the exceptional pair excludes cyclic permutation of three characters modulo a common twist.

For a nonzero primitive class $v\in H^{p,q}$ of degree $k=p+q$ on a compact Kähler manifold of complex dimension $d$, the displayed expression is positive. Together with orthogonality of complementary Hodge bidegrees, this makes primitive cohomology a polarized Hodge structure when the Kähler class is rational, after clearing denominators. Positivity on the full cohomology requires the signs prescribed by its primitive decomposition.

By positive unfolding of a cusp-form square, $\sum |a_n{|}^2{n}^{-\beta }<\infty$ for $\beta >k$. Choose $k<\beta <2\sigma -1$. The Cauchy-Schwarz inequality bounds the absolute sum for the L-function of a cusp form by

On a compact Kähler manifold $X$, if $P$ is a positive holomorphic line bundle, then $H^i(X,K_X\otimes P)=0$ for $i>0$. The canonical-bundle twist is part of the hypothesis and conclusion; positivity alone does not remove it on a general manifold. For a complex torus, the canonical bundle is trivial, so higher cohomology of a positive line bundle vanishes without an additional twist.