Highest-weight representation

A highest-weight representation is generated by a weight vector killed by every positive-root subspace.

The generating vector has a highest Weight relative to the chosen positive root system.

For a complex semisimple Lie algebra and a choice of positive roots, every finite-dimensional Irreducible Lie algebra representation has a unique highest weight, which is a dominant integral weight. Conversely each dominant integral weight labels exactly one such irreducible module up to isomorphism. Its highest-weight vector spans the highest-weight space and generates the module. This classification and Weyl complete reducibility theorem reduce decomposing any finite-dimensional module to its irreducible highest-weight summands.

The A2 root system representation $R(2,0)={\mathrm{Sym}}^2{\mathbb{C}}^3$ has dimension six and weights $(2,0),(0,1),(1,-1),(-2,2),(-1,0),(0,-2)$ in Dynkin label coordinates, each with multiplicity one.

The tensor product of Lie algebra representations decomposes as $R(2,0)\otimes R(2,0)=R(4,0)\oplus R(2,1)\oplus R(0,2)$, of dimensions $15,15,6$. The exterior square is $R(2,1)$ and the symmetric square is $R(4,0)\oplus R(0,2)$. Highest-weight vectors and the dimension formula determine the decomposition.

The tensor square has fifteen distinct weights: three outer corners of multiplicity one, six near-corner edge weights of multiplicity two, three edge midpoints of multiplicity three, and three interior weights of multiplicity four. In the irreducible components only $R(2,1)$ is weight-degenerate: $(1,0),(-1,1),(0,-1)$ each have multiplicity two. All weights of $R(4,0)$ and $R(0,2)$ have multiplicity one.

If a dominant integral weight $\lambda$ lies in the root lattice, its finite-dimensional irreducible highest-weight representation contains zero as a weight. First $\lambda$ is a nonnegative integral combination of simple roots: a negative part would have negative inner product with $\lambda$, contradicting dominance. For any nonzero weight $\mu =\sum m_i{\alpha }_i$ with $m_i\ge 0$, the identity $(\mu ,\mu )=\sum m_i(\mu ,{\alpha }_i)>0$ supplies an index with $m_i>0$ and $⟨\mu ,{\alpha }_i^{\vee }⟩>0$. By Injectivity of sl2 lowering above weight zero, applying the corresponding lowering operator produces a nonzero vector of weight $\mu -{\alpha }_i$. Repetition reaches zero.

A finite-dimensional irreducible representation of a complex semisimple Lie algebra is minuscule if its weights form one Weyl group orbit. Every weight multiplicity is then one, because multiplicities are invariant under the Weyl group and the highest weight has multiplicity one.

The formal character of a weight module for a minuscule representation of highest weight $\lambda$ is $\mathrm{ch}V(\lambda )={\sum }_{\mu \in W\lambda }{e}^{\mu }$. The sum is over distinct points of the Weyl group orbit, so $\dim V(\lambda )=|W\lambda |$.

A dominant integral weight $\lambda$ is minuscule when its irreducible highest-weight representation is a minuscule representation. Equivalently, $⟨\lambda ,{\alpha }^{\vee }⟩\in \{0,1\}$ for every positive root $\alpha$. This convention includes the zero weight and its trivial Lie algebra representation.

By dominant root-lattice highest weights have zero weight, a minuscule weight in the root lattice has zero in its Weyl group orbit. Since every element of the Weyl group is invertible, the highest weight must itself be zero. Thus equality of the weight lattice and root lattice rules out every nontrivial minuscule representation without requiring a classification of root systems.

The highest weight of a highest-weight representation is the weight of a generating vector annihilated by every positive-root operator. Finite-dimensional irreducible representations of a complex semisimple Lie algebra are classified by dominant integral highest weights.

The coordinates of a highest weight record its eigenvalues on a chosen basis of a Cartan subalgebra. They depend on that basis. Dynkin labels instead use the simple coroots; orthonormal Cartesian coordinates and Dynkin labels must be converted before applying a dimension or Casimir eigenvalue formula.

With $H_1=\mathrm{diag}(1,-1,0)/2$ and $H_2=\mathrm{diag}(1,1,-2)/(2\sqrt{3})$, the simple roots are $(1,0)$ and $(-1/2,\sqrt{3}/2)$. Their coroots give Dynkin labels $a=2p$ and $b=\sqrt{3}q-p$ for Cartesian highest-weight coordinates $(p,q)$. Finite-dimensional irreducible representations require $a,b\in {\mathbb{Z}}_{\ge 0}$. The vertical coordinate is $\sqrt{3}/2$ times flavor hypercharge.

A ladder operator maps weight vectors to weight vectors whose weights differ by one root. Positive-root vectors raise weights and negative-root vectors lower them.

A raising operator is a positive-root vector. Acting on a weight vector of weight $\mu$ either gives zero or a weight vector of weight $\mu +\alpha$.

A lowering operator is a negative-root vector. Acting on a weight vector of weight $\mu$ either gives zero or a weight vector of weight $\mu -\alpha$.

A Cartan-Weyl basis consists of a basis of a Cartan subalgebra together with root vectors $E_{\alpha }$. It makes weight raising and lowering explicit through $[H,E_{\alpha }]=\alpha (H)E_{\alpha }$.

A Chevalley basis imposes an additional integral normalization on these root vectors; that stricter basis is not interchangeable with an arbitrary Cartan-Weyl basis.

A disentangling identity factors the exponential of a sum of noncommuting Lie-algebra generators into an ordered product of exponentials. For ${\mathfrak{sl}}_2$, a common order is $e^{a{E}_{-}}{e}^{bH}{e}^{c{E}_{+}}$.

Every finite-dimensional representation of a complex semisimple Lie algebra is a direct sum of irreducible representations.

If $\lambda ={\sum }_i{m}_i{\omega }_i$ is dominant integral, the tensor product ${⨂}_{i}V({\omega }_i{)}^{\otimes m_i}$ contains $V(\lambda )$ as the irreducible summand generated by the tensor product of highest-weight vectors.

If $\mu$ and $\lambda$ are weights of a finite-dimensional representation, $\mu$ is dominant, and $\mu ⪯\lambda$, thenSuccessively subtract a simple root ${\alpha }_i$ for which $⟨\lambda ,{\alpha }_i^{\vee }⟩>0$ and use injectivity of the corresponding ${\mathfrak{sl}}_2$ lowering operator.

A vector $v$ has weight $\mu \in {\mathfrak{t}}^{\ast }$ when $hv=\mu (h)v$ for every $h$ in the Cartan subalgebra.

A Weight is the eigenvalue functional; a weight vector is a nonzero vector carrying it.

A highest-weight vector is a nonzero weight vector annihilated by every positive-root raising operator. The module it generates is a highest-weight representation.

A weight of a representation is a functional $\mu$ whose weight space is nonzero.

The weight space $V_{\mu }$ consists of all weight vectors of weight $\mu$, together with zero.

This is the simultaneous eigenspace belonging to a Weight.

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.

For a finite-dimensional complex Lie algebra representation $\rho$ of a nilpotent Lie algebra $H$, a generalized weight space is the simultaneous generalized eigenspace displayed above, where $\lambda :H\to \mathbb{C}$ is a linear functional. Every occurring $\lambda$ vanishes on the derived algebra of $H$. Ordinary weight vectors satisfy the same equations without the powers.

Every finite-dimensional complex Lie algebra representation of a nilpotent Lie algebra decomposes into invariant generalized weight spaces. The nilpotence assumption is essential: a general solvable action can have extensions between distinct weights.

Here is a proof. For $h,k\in H$, nilpotence gives $(\mathrm{ad}h{)}^{N}k=0$, hence $(\mathrm{ad}\rho (h){)}^N\rho (k)=0$. By the nilpotent commutator preserves generalized eigenspaces lemma, every generalized eigenspace of $\rho (h)$ is invariant under every $\rho (k)$. Split successively by the acting operators of a basis of $H$. Each resulting block is invariant under $H$, and each basis operator has only one eigenvalue on it. The Lie theorem triangularizes the action on each block; its diagonal entries therefore give one and the same linear functional $\lambda$ on every basis vector of that block. Every $\rho (h)-\lambda (h)I$ is strictly upper triangular there, proving that the block is $V^{\lambda }$. Distinct blocks have distinct weights, so this is a direct sum. On the zero generalized weight space, every acting operator is a nilpotent endomorphism, so the Engel theorem supplies a nonzero common annihilated vector whenever that space is nonzero.

Let $H$ be a Cartan subalgebra of a finite-dimensional complex Lie algebra $L$. In the adjoint generalized-weight decomposition for a nilpotent Lie algebra, its zero summand satisfies $L^0=H$. Nilpotence gives $H\subseteq L^0$. If $L^0/H$ were nonzero, all adjoint operators of $H$ on this quotient would be nilpotent, so the Engel theorem would supply a nonzero coset $x+H$ with $[H,x]\subseteq H$. That contradicts self-normalization. The derivation rule ensures that $L^0$ is a subalgebra, so the quotient action is well-defined.

For a finite-dimensional representation of a complex semisimple Lie algebra, the weights on a line $\mu +\mathbb{Z}\alpha$ form a contiguous string from $\mu +p\alpha$ to $\mu -q\alpha$, with $q-p=⟨\mu ,{\alpha }^{\vee }⟩$. Restriction to the associated sl2 subalgebra gives nested strings of the same parity, so the string endpoints determine the weight set but not its multiplicities.

Start with the highest weight and process weights by increasing height below it. For each simple root, compute the upper endpoint from the already known higher weights, use the weight string endpoint relation, and append all downward steps. Repeat until the finite set stabilizes. This recovers the weights of a finite-dimensional Irreducible Lie algebra representation without determining weight multiplicities.

A weight-space decomposition expresses a Lie algebra representation as a direct sum of simultaneous eigenspaces for a Cartan subalgebra. Each summand is the weight space for its weight. In complex finite-dimensional representations of a semisimple Lie algebra, the simple coroots act as commuting diagonalizable operators, giving this decomposition.

A weight basis of a Lie algebra representation is a basis consisting of weight vectors for a chosen Cartan subalgebra. When the Cartan action is diagonalizable, choose a basis separately in every weight space and combine them. This makes weight multiplicities directly visible and gives diagonal matrices for Cartan elements.

A diagram placing the weights of a representation in the real span of its weight lattice, recording each weight multiplicity. Coincident weights must retain their multiplicity: for an Adjoint representation the zero-weight multiplicity is the dimension of its Cartan subalgebra.

Let $P_{a,b}$ be the convex hull of the Weyl group orbit of the highest weight $a{\omega }_1+b{\omega }_2$ for the SU(3) Lie algebra. Restrict the displayed formula to the weight lattice coset $p{\omega }_1+q{\omega }_2+$ root lattice. It describes successive inward shells: the outer boundary has weight multiplicity one, each inward shell increases it by one, and the final triangular region has the constant value $\min (p,q)+1$. When $p=q$ that region is a single point. A triangular diagram with $p=0$ or $q=0$ has multiplicity one throughout. This is a geometric form of the dominant weight multiplicity formula for sl3, transported by Weyl reflections.

// Target: lie-theory.bigb

Using $I_3=\mathrm{diag}(1/2,-1/2,0)$ and flavour $Y=\mathrm{diag}(1/3,1/3,-2/3)$, the three coordinate vectors of the SU(3) group defining group representation have the displayed weights of a representation. The complex-conjugate group representation has their negatives. These coordinates use flavor hypercharge; another Cartan normalization rescales the vertical axis.

For a tensor product of Lie algebra representations with weight-space decompositions, the weights add and their weight multiplicities convolve. A diagram must sum contributions landing at the same weight, rather than treating coincident points as distinct positions.

For a finite-dimensional irreducible highest-weight representation with highest weight $\lambda$, every weight in the Weyl group orbit $W\lambda$ is extremal and has weight multiplicity one.

The multiplicity of a weight $\mu$ is the dimension of its weight space.

Thus it is a dimension attached to a Weight and its weight space.

Multiply the Weyl character formula by the generating series of the Kostant partition function. In each numerator summand, the coefficient of $e^{\mu }$ occurs when $\nu =w(\lambda +\rho )-(\mu +\rho )$. Summing those coefficients with the Weyl group signs proves the displayed weight multiplicity formula. Although $\rho$ need not itself be an integral group weight, $w\rho -\rho$ is in the root lattice, so each partition-function argument is an integral weight.

For linear maps $E:U\to W$ and $F:W\to U$ between finite-dimensional vector spaces, ${\mathrm{tr}}_U(FE)={\mathrm{tr}}_W(EF)$ even when their dimensions differ; expanding rectangular matrices proves it. For root vectors normalized by $[E_{\alpha },F_{\alpha }]=t_{\alpha }$, with $B(t_{\alpha },h)=\alpha (h)$, this gives $T_{\alpha }(\mu )=T_{\alpha }(\mu +\alpha )+(\mu +\alpha ,\alpha )\dim V_{\mu +\alpha }$. Iteration along a finite weight string is the trace step in Freudenthal multiplicity formula.

For a finite-dimensional Irreducible Lie algebra representation of a complex semisimple Lie algebra with highest weight $\lambda$, this recursion computes its weight multiplicities from $m_{\lambda }(\lambda )=1$. Here $\rho$ is the half-sum of positive roots and the inner product is induced by the Killing form. Write the Casimir operator on the weight-$\mu$ space as $(\mu ,\mu +2\rho )I+2{\sum }_{\alpha >0}{F}_{\alpha }{E}_{\alpha }$. The cyclic trace identity between adjacent weight spaces gives $\mathrm{tr}(F_{\alpha }{E}_{\alpha }{|}_{V_{\mu }})={\sum }_{j\ge 1}(\mu +j\alpha ,\alpha )m_{\lambda }(\mu +j\alpha )$. Taking the trace and using the Casimir eigenvalue proves the formula. Only finitely many terms are nonzero.

A singular vector is a nonzero weight vector annihilated by the positive nilpotent subalgebra ${\mathfrak{n}}^{+}$.

For a Borel subalgebra $\mathfrak{b}=\mathfrak{t}\oplus {\mathfrak{n}}^{+}$ and a weight $\lambda$, the Verma module iswhere ${\mathfrak{n}}^{+}$ kills ${\mathbb{C}}_{\lambda }$ and $h\in \mathfrak{t}$ acts by $\lambda (h)$.

If a $\mathfrak{g}$-module contains a vector $v$ of weight $\lambda$ annihilated by ${\mathfrak{n}}^{+}$, there is a unique homomorphism $M_{\lambda }\to V$ sending the canonical highest-weight vector to $v$. Every highest-weight module of weight $\lambda$ is therefore a quotient of $M_{\lambda }$.

The sum of all proper submodules of $M_{\lambda }$ is its unique maximal proper submodule. The quotient is the unique irreducible highest-weight module $V(\lambda )$.

If $\lambda$ is a dominant integral weight, the maximal proper submodule of $M(\lambda )$ is generated by the simple-root singular vectorsThe generator associated with ${\alpha }_i$ has weight $s_i(\lambda +\rho )-\rho$.

For a module with finite-dimensional weight spaces, its formal character is $\mathrm{ch}V={\sum }_{\mu }(\dim V_{\mu })e^{\mu }$ in the completed group algebra of the weight lattice.

For a finite-dimensional representation of a complex semisimple Lie algebra, select a maximal remaining dominant weight, subtract that multiplicity of the corresponding irreducible formal character, and repeat. The Weyl complete reducibility theorem guarantees nonnegative residual multiplicities and termination. Checking the full character or the exhausted total dimension confirms that no summands remain.

For the special linear Lie algebra ${\mathfrak{sl}}_3$, the irreducible highest-weight representation with Dynkin labels $(a,b)$ has the displayed formal character on diagonal matrices with $x_1{x}_2{x}_3=1$. The interlacing rows $(a+b,b,0)$, $(p,q)$ and $(r)$ enumerate a Gelfand–Tsetlin basis; each triple contributes the exponent differences $(r-(p+q-r),(p+q-r)-(a+2b-p-q))$ as its weight. This gives exact integer weight multiplicities, including repeated weights, without floating-point character evaluation.

For the irreducible highest-weight representation of the special linear Lie algebra ${\mathfrak{sl}}_3$ with Dynkin labels $(a,b)$, let $(p,q)$ be a dominant integral weight. Put $r=(2a+b-2p-q)/3$ and $s=(a+2b-p-2q)/3$. If these are not integers, the weight multiplicity is zero; otherwise it is $\max \{0,1+\min (a,b,r,s)\}$.

To derive the formula, use the sl3 interlacing character formula with top row $(a+b,b,0)$. The target diagonal exponents are $(s+p+q,s+q,s)$, and an interlacing middle row $(P,Q)$ has $P+Q=2s+p+2q$, with bottom entry $s+p+q$. ThusEvery integral $P$ in this interval gives exactly one Gelfand–Tsetlin basis vector. The upper bound minus the lower bound is the minimum of six differences: $a,s,r,r+p+q,b,s+q$. Since $p,q\ge 0$, this minimum reduces to $\min (a,b,r,s)$. Counting the interval proves the formula, including absent weights of a representation. This converts a two-dimensional weight diagram calculation into four elementary boundary distances.

For a dominant integral weight $\lambda$, the irreducible character is

For a Weyl group acting on a weight lattice, the displayed element of its group ring is alternating: applying $s\in W$ multiplies it by $\det (s)$, by reindexing the sum. It vanishes when $\nu$ is fixed by a root reflection. For the Weyl vector $\rho$, it is the Weyl denominator. The Weyl character formula is the ratio $A_{\lambda +\rho }/A_{\rho }$. These general root-system alternants specialize to monomial determinants in type A but are not restricted to that type.

For strictly decreasing integer tuples $\kappa ,\eta$, let $A_{\kappa }(z)=\det (z_i^{{\kappa }_j})$ on the diagonal maximal torus of the unitary group. Expanding both determinants, Fourier orthogonality makes the integral zero unless the exponent tuples have the same permutation orbit. Strict decreasing order makes that equivalent to $\kappa =\eta$; the $n!$ matching permutation terms then each contribute one. An alternating Laurent polynomial is a finite linear combination of these alternants. For an irreducible character ${\chi }_{\lambda }$, its highest monomial gives coefficient one for $A_{\lambda +\delta }$ in ${\chi }_{\lambda }{A}_{\delta }$, with $\delta =(n-1,\dots ,0)$. The Weyl integration formula and Schur orthogonality relations force the sum of squared coefficients to be one, so all other coefficients vanish. This proves the unitary Weyl character formula directly.

For a dominant integer tuple $\lambda$ and $\delta =(m-1,\dots ,0)$, the rational Schur character is $\det (x_i^{{\lambda }_j+m-j})/\det (x_i^{m-j})$. For partitions it is a symmetric polynomial; for general tuples it is Laurent on the diagonal torus. The formula follows by comparing the trace of a permuted tensor power with the Frobenius alternant character formula and using independence of symmetric-group characters. Apparent poles at coincident nonzero eigenvalues are removable.

For a complex semisimple Lie algebra and an integer $k\ge 0$, the Weyl character formula and Weyl denominator formula giveApply the dilation $e^{\mu }\mapsto e^{(k+1)\mu }$ to the denominator identity to obtain the numerator. Cancellation leaves a finite geometric series for each positive root of a root system. Evaluating the resulting polynomial at all formal exponentials equal to one gives the dimension, without taking an undefined quotient at the identity.

For a positive root system ${\Phi }^{+}$,

The Weyl denominator is the alternating expression in the Weyl denominator formula. Its product form uses the Weyl vector $\rho$, half the positive-root sum. On a compact maximal torus, the factor $e^{\rho }$ has unit modulus; if it is only defined on a covering torus, the squared modulus still descends. Thus $|\Delta (t){|}^2={\prod }_{\alpha >0}|1-\alpha (t){|}^2$ is a globally defined density in the Weyl integration formula. For the unitary group an integer central shift gives the Vandermonde determinant $\det (z_i^{n-j})$, whose modulus is the same.

The dimension obtained by evaluating the Weyl character formula at the identity is

For a root system with $N$ positive roots, put $A_{\nu }(H)={\sum }_{w\in W}\det (w)e^{(w\nu ,H)}$. Its degree-$m$ Taylor coefficient is alternating separately in $\nu$ and $H$, and hence divisible in each set of variables by the product of positive-root linear forms. Thus it vanishes for $m<N$ and its degree-$N$ term is $C{\prod }_{\alpha >0}(\nu ,\alpha ){\prod }_{\alpha >0}(H,\alpha )$. The Weyl denominator formula at $\nu =\rho$ determines the nonzero constant $C$. Taking the ratio $A_{\lambda +\rho }(tH)/A_{\rho }(tH)$ for regular $H$ and letting $t\to 0$ gives the Weyl dimension formula $\dim L(\lambda )={\prod }_{\alpha >0}(\lambda +\rho ,\alpha )/(\rho ,\alpha )$. Factors converting roots to coroots cancel.

For the G2 root system with ${\alpha }_1$ short and ${\alpha }_2$ long, the fundamental weights are ${\omega }_1=2{\alpha }_1+{\alpha }_2$ and ${\omega }_2=3{\alpha }_1+2{\alpha }_2$. The positive coroots, relative to the simple coroots, have coordinates $(1,0),(0,1),(1,3),(2,3),(1,1),(1,2)$. The Weyl dimension formula therefore gives, for nonnegative integers $a,b$,The denominator is the product of the pairings of the Weyl vector with those coroots. The dimensions at ${\omega }_1,{\omega }_2,2{\omega }_1$ are seven, fourteen and twenty-seven.

For the C2 root system with its first simple root short, the irreducible highest-weight representation of highest weight $a{\omega }_1+b{\omega }_2$, where $a,b\in {\mathbb{Z}}_{\ge 0}$, has dimension

For the principal specialization, let $h_{\mathrm{pr}}=2{\rho }^{\vee }$, so every simple root takes value two on $h_{\mathrm{pr}}$. The q-character isIt is obtained from the Weyl character formula by substituting $e^{\mu }=q^{\mu (h_{\mathrm{pr}})}$.

Use ${\mathrm{ch}}_{q}V={\mathrm{tr}}_V{q}^{2{\rho }^{\vee }}$. The positive roots of a root system of the G2 root system have simple-root heights $1,1,2,3,4,5$. Each root space has dimension one, and the zero weight space has dimension two. ConsequentlyBy the classification of finite-dimensional sl2 representations, restriction to the Principal sl2 subalgebra is $L_{10}\oplus L_2$, of dimensions eleven and three. If the grading uses ${\rho }^{\vee }$ instead, divide every exponent by two.

A crystal basis is the combinatorial limit of a basis of a quantum-group representation. Its colored directed graph records the actions of the Kashiwara lowering operators.

With long-first simple roots ${\alpha }_1={\epsilon }_1-{\epsilon }_2$, ${\alpha }_2={\epsilon }_2$, the Adjoint representation has highest weight ${\epsilon }_1+{\epsilon }_2=2{\Lambda }_2$. Its crystal basis has one vertex $x_{\beta }$ for every root and two distinct zero-weight vertices $z_1,z_2$. The color-one strings are $x_{{\epsilon }_1}\to x_{{\epsilon }_2}$, $x_{{\alpha }_1}\to z_1\to x_{-{\alpha }_1}$, and $x_{-{\epsilon }_2}\to x_{-{\epsilon }_1}$. The color-two strings are $x_{{\epsilon }_1+{\epsilon }_2}\to x_{{\epsilon }_1}\to x_{{\epsilon }_1-{\epsilon }_2}$, $x_{{\epsilon }_2}\to z_2\to x_{-{\epsilon }_2}$, and $x_{-{\epsilon }_1+{\epsilon }_2}\to x_{-{\epsilon }_1}\to x_{-{\epsilon }_1-{\epsilon }_2}$. All other edges vanish. One construction takes the component generated by $1\otimes 2$ in the tensor square of the crystal of the defining odd-orthogonal representation; the crystal tensor-product rule produces exactly these ten vertices. The equivariant isomorphism ${\Lambda }^2{\mathbb{C}}^5\cong {\mathfrak{so}}_5$ identifies its highest weight and representation.

For the B2 root system with ${\alpha }_1={\epsilon }_1-{\epsilon }_2$ long and ${\alpha }_2={\epsilon }_2$ short, the spin fundamental weight is ${\Lambda }_2=({\epsilon }_1+{\epsilon }_2)/2$. Its four-dimensional highest-weight representation has the minuscule weight orbit $\frac{1}{2}(\pm {\epsilon }_1\pm {\epsilon }_2)$, each of multiplicity one. The crystal basis is the chain $++\stackrel{2}{\to }+-\stackrel{1}{\to }-+\stackrel{2}{\to }--$, where signs denote the coordinates. Each Kashiwara operator subtracts its indicated simple root; the sl2 Lie algebra strings have length one. This fixes every edge and gives a connected highest-weight graph with the correct character.

For the symplectic Lie algebra ${\mathfrak{sp}}_{2n}$, the defining crystal basis is the chain $1,2,\dots ,n,\bar{n},\dots ,\bar{1}$, with weights ${\epsilon }_1,\dots ,{\epsilon }_n,-{\epsilon }_n,\dots ,-{\epsilon }_1$. Its successive lowering-edge colors are $1,2,\dots ,n-1,n,n-1,\dots ,1$. Every step subtracts its color's simple root. Applying the crystal tensor-product rule to two copies gives a complete grid description of the tensor-square crystal. For $n\ge 2$, its only highest vertices are $1\otimes 1$, $1\otimes 2$, and $1\otimes \bar{1}$, of highest weights $2{\omega }_1$, ${\omega }_2$, and zero. For $n=1$, the middle vertex is absent and only the other two highest vertices remain.

Under the crystal tensor-product rule that uses the first factor for lowering when ${\phi }_i(a)>{\epsilon }_i(b)$, the four-letter C2 root system crystal has highest tensor vertices $1\otimes 1$, $1\otimes 2$ and $1\otimes \bar{1}$. Their highest weights are $2{\omega }_1$, ${\omega }_2$ and zero. Lowering generates connected components of sizes ten, five and one, respectively; their union exhausts all sixteen vertices. The corresponding representation decomposition is the symmetric square plus the primitive exterior square and the invariant symplectic line.

The seven-dimensional irreducible representation of the G2 root system has highest weight ${\omega }_1=2{\alpha }_1+{\alpha }_2$. Its crystal basis is the chain of weights $2{\alpha }_1+{\alpha }_2,{\alpha }_1+{\alpha }_2,{\alpha }_1,0,-{\alpha }_1,-{\alpha }_1-{\alpha }_2,-2{\alpha }_1-{\alpha }_2$, with successive edge colors $1,2,1,1,2,1$. Each step subtracts its color's simple root. The nonzero weights are the six short roots, each of multiplicity one; the zero weight also has multiplicity one.

For $V=L({\omega }_1)$ with the G2 root system numbered short-first,In the chain basis of the crystal of the seven-dimensional G2 representation, normalize $v_1=f_1{v}_0$, $v_2=f_2{v}_1$, $v_3=f_1{v}_2$. Then $v_0\wedge v_1$ and $v_0\wedge v_3-2v_1\wedge v_2$ are nonzero highest-weight vectors of weights ${\omega }_2$ and ${\omega }_1$; the identities $e_1{v}_1=v_0$, $e_1{v}_3=2v_2$, $e_2{v}_2=v_1$ verify both raising conditions. The Weyl complete reducibility theorem and G2 dimension polynomial exhaust the twenty-one dimensions of the exterior square. In the symmetric square, $v_0^2$ generates the twenty-seven-dimensional summand. Self-duality supplies a Lie-invariant bilinear form, which is symmetric because a nondegenerate alternating bilinear form cannot have odd dimension; its inverse gives the remaining invariant line.

The defining representation of ${\mathfrak{so}}_{2n+1}$ has weights ${\epsilon }_1,\dots ,{\epsilon }_n,0,-{\epsilon }_n,\dots ,-{\epsilon }_1$. Its crystal basis is a chain on these vertices, with lowering-edge colors $1,\dots ,n-1,n,n,n-1,\dots ,1$. The short-root string has length two through zero. With the crystal tensor-product rule, for $n\ge 2$ the tensor square has highest vertices $1\otimes 1$, $1\otimes 2$ and $1\otimes \bar{1}$, corresponding to the traceless symmetric square, the exterior square and the trivial Lie algebra representation. Their dimensions are $(2n+1)(n+1)-1$, $n(2n+1)$ and one.

For each simple root ${\alpha }_i$, the $i$-colored edges of a crystal form strings. Along such a string, ${\tilde{e}}_i$ raises the weight by ${\alpha }_i$ and ${\tilde{f}}_i$ lowers it by ${\alpha }_i$.

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 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.

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 ${\sum }_i{e}_i\otimes e_i^{\ast }$.

Write ${\epsilon }_i(b)$ and ${\phi }_i(b)$ for the numbers of raising and lowering steps of color $i$ available from a vertex of a crystal basis. One convention for the tensor product of crystals isThe raising operator uses the first factor for ${\phi }_i(a)\ge {\epsilon }_i(b)$ and the second otherwise. The different strictness makes raising and lowering inverse along every edge. An edge is absent if the chosen Kashiwara operator vanishes. Stating the convention is essential when identifying the highest-weight vectors as ordered tensor vertices.