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Weyl character formula
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<h2 id="statement-of-weyl-character-formula">Statement of Weyl character formula</h2> <p>The character formula can be expressed in terms of representations of complex semisimple Lie algebras or in terms of the (essentially equivalent) representation theory of compact <a href="/facts/Lie_group/a9jCQyjF">Lie groups</a>. </p> <h3>Complex semisimple Lie algebras</h3> <p>Let π {\displaystyle \pi } be an irreducible, finite-dimensional representation of a complex <a href="/facts/Semisimple_Lie_algebra/m8jA3Bcf">semisimple Lie algebra</a> g {\displaystyle {\mathfrak {g}}} . Suppose h {\displaystyle {\mathfrak {h}}} is a <a href="/facts/Cartan_subalgebra/8gymnPU0">Cartan subalgebra</a> of g {\displaystyle {\mathfrak {g}}} . The character of π {\displaystyle \pi } is then the function ch π : h → C {\displaystyle \operatorname {ch} _{\pi }:{\mathfrak {h}}\rightarrow \mathbb {C} } defined by </p> ch π ( H ) = tr ( e π ( H ) ) . {\displaystyle \operatorname {ch} _{\pi }(H)=\operatorname {tr} (e^{\pi (H)}).} <p>The value of the character at H = 0 {\displaystyle H=0} is the dimension of π {\displaystyle \pi } . By elementary considerations, the character may be computed as </p> ch π ( H ) = ∑ μ m μ e μ ( H ) {\displaystyle \operatorname {ch} _{\pi }(H)=\sum _{\mu }m_{\mu }e^{\mu (H)}} , <p>where the sum ranges over all the <a href="/facts/Weight_(representation_theory)/7N8q5sw0">weights</a> μ {\displaystyle \mu } of π {\displaystyle \pi } and where m μ {\displaystyle m_{\mu }} is the multiplicity of μ {\displaystyle \mu } . (The preceding expression is sometimes taken as the definition of the character.) </p><p>The character formula states<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> that ch π ( H ) {\displaystyle \operatorname {ch} _{\pi }(H)} may also be computed as </p> ch π ( H ) = ∑ w ∈ W ε ( w ) e w ( λ + ρ ) ( H ) ∏ α ∈ Δ + ( e α ( H ) / 2 − e − α ( H ) / 2 ) {\displaystyle \operatorname {ch} _{\pi }(H)={\frac {\sum _{w\in W}\varepsilon (w)e^{w(\lambda +\rho )(H)}}{\prod _{\alpha \in \Delta ^{+}}(e^{\alpha (H)/2}-e^{-\alpha (H)/2})}}} <p>where </p> <ul><li> W {\displaystyle W} is the <a href="/facts/Weyl_group/N9QG4Wyn">Weyl group</a>;</li> <li> Δ + {\displaystyle \Delta ^{+}} is the set of the <a href="/facts/Positive_root/TFaEdwMU">positive roots</a> of the <a href="/facts/Root_system/TFaEdwMU">root system</a> Δ {\displaystyle \Delta } ;</li> <li> ρ {\displaystyle \rho } is the half-sum of the positive roots, often called the <i>Weyl vector</i>;</li> <li> λ {\displaystyle \lambda } is the <a href="/facts/Highest_weight/7N8q5sw0">highest weight</a> of the irreducible representation π {\displaystyle \pi } ;</li> <li> ε ( w ) {\displaystyle \varepsilon (w)} is the determinant of the action of w {\displaystyle w} on the <a href="/facts/Cartan_subalgebra/8gymnPU0">Cartan subalgebra</a> h ⊂ g {\displaystyle {\mathfrak {h}}\subset {\mathfrak {g}}} . This is equal to ( − 1 ) ℓ ( w ) {\displaystyle (-1)^{\ell (w)}} , where ℓ ( w ) {\displaystyle \ell (w)} is the <a href="/facts/Weyl_group/N9QG4Wyn">length of the Weyl group element</a>, defined to be the minimal number of reflections with respect to simple roots such that w {\displaystyle w} equals the product of those reflections.</li></ul> <h3>Discussion</h3> <p>Using the Weyl denominator formula (described below), the character formula may be rewritten as </p> ch π ( H ) = ∑ w ∈ W ε ( w ) e w ( λ + ρ ) ( H ) ∑ w ∈ W ε ( w ) e w ( ρ ) ( H ) {\displaystyle \operatorname {ch} _{\pi }(H)={\frac {\sum _{w\in W}\varepsilon (w)e^{w(\lambda +\rho )(H)}}{\sum _{w\in W}\varepsilon (w)e^{w(\rho )(H)}}}} , <p>or, equivalently, </p> ch π ( H ) ∑ w ∈ W ε ( w ) e w ( ρ ) ( H ) = ∑ w ∈ W ε ( w ) e w ( λ + ρ ) ( H ) . {\displaystyle \operatorname {ch} _{\pi }(H){\sum _{w\in W}\varepsilon (w)e^{w(\rho )(H)}}=\sum _{w\in W}\varepsilon (w)e^{w(\lambda +\rho )(H)}.} <p>The character is itself a large sum of exponentials. In this last expression, we then multiply the character by an alternating sum of exponentials—which seemingly will result in an even larger sum of exponentials. The surprising part of the character formula is that when we compute this product, only a small number of terms actually remain. Many more terms than this occur at least once in the product of the character and the Weyl denominator, but most of these terms cancel out to zero.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> The only terms that survive are the terms that occur only once, namely e ( λ + ρ ) ( H ) {\displaystyle e^{(\lambda +\rho )(H)}} (which is obtained by taking the highest weight from ch π {\displaystyle \operatorname {ch} _{\pi }} and the highest weight from the Weyl denominator) and things in the Weyl-group orbit of e ( λ + ρ ) ( H ) {\displaystyle e^{(\lambda +\rho )(H)}} . </p> <h3>Compact Lie groups</h3> <p>Let K {\displaystyle K} be a compact, connected Lie group and let T {\displaystyle T} be a maximal torus in K {\displaystyle K} . Let Π {\displaystyle \Pi } be an irreducible representation of K {\displaystyle K} . Then we define the character of Π {\displaystyle \Pi } to be the function </p> X ( x ) = trace ( Π ( x ) ) , x ∈ K . {\displaystyle \mathrm {X} (x)=\operatorname {trace} (\Pi (x)),\quad x\in K.} <p>The character is easily seen to be a class function on K {\displaystyle K} and the <a href="/facts/Peter%25E2%2580%2593Weyl_theorem/vQfn6cQr">Peter–Weyl theorem</a> asserts that the characters form an orthonormal basis for the space of square-integrable class functions on K {\displaystyle K} .<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> </p><p>Since X {\displaystyle \mathrm {X} } is a class function, it is determined by its restriction to T {\displaystyle T} . Now, for H {\displaystyle H} in the Lie algebra t {\displaystyle {\mathfrak {t}}} of T {\displaystyle T} , we have </p> trace ( Π ( e H ) ) = trace ( e π ( H ) ) {\displaystyle \operatorname {trace} (\Pi (e^{H}))=\operatorname {trace} (e^{\pi (H)})} , <p>where π {\displaystyle \pi } is the associated representation of the Lie algebra k {\displaystyle {\mathfrak {k}}} of K {\displaystyle K} . Thus, the function H ↦ trace ( Π ( e H ) ) {\displaystyle H\mapsto \operatorname {trace} (\Pi (e^{H}))} is simply the character of the associated representation π {\displaystyle \pi } of k {\displaystyle {\mathfrak {k}}} , as described in the previous subsection. The restriction of the character of Π {\displaystyle \Pi } to T {\displaystyle T} is then given by the same formula as in the Lie algebra case: </p> X ( e H ) = ∑ w ∈ W ε ( w ) e w ( λ + ρ ) ( H ) ∑ w ∈ W ε ( w ) e w ( ρ ) ( H ) . {\displaystyle \mathrm {X} (e^{H})={\frac {\sum _{w\in W}\varepsilon (w)e^{w(\lambda +\rho )(H)}}{\sum _{w\in W}\varepsilon (w)e^{w(\rho )(H)}}}.} <p>Weyl's <i>proof</i> of the character formula in the compact group setting is completely different from the algebraic proof of the character formula in the setting of semisimple Lie algebras.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> In the compact group setting, it is common to use "real roots" and "real weights", which differ by a factor of i {\displaystyle i} from the roots and weights used here. Thus, the formula in the compact group setting has factors of i {\displaystyle i} in the exponent throughout. </p> <h3>The SU(2) case</h3> <p>In the case of the group SU(2), consider the <a href="/facts/Representation_theory_of_SU(2)/fBw5fr2r">irreducible representation</a> of dimension m + 1 {\displaystyle m+1} . If we take T {\displaystyle T} to be the diagonal subgroup of SU(2), the character formula in this case reads<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> </p> X ( ( e i θ 0 0 e − i θ ) ) = e i ( m + 1 ) θ − e − i ( m + 1 ) θ e i θ − e − i θ = sin ( ( m + 1 ) θ ) sin θ . {\displaystyle \mathrm {X} \left({\begin{pmatrix}e^{i\theta }&0\\0&e^{-i\theta }\end{pmatrix}}\right)={\frac {e^{i(m+1)\theta }-e^{-i(m+1)\theta }}{e^{i\theta }-e^{-i\theta }}}={\frac {\sin((m+1)\theta )}{\sin \theta }}.} <p>(Both numerator and denominator in the character formula have two terms.) It is instructive to verify this formula directly in this case, so that we can observe the cancellation phenomenon implicit in the Weyl character formula. </p><p>Since the representations are known very explicitly, the character of the representation can be written down as </p> X ( ( e i θ 0 0 e − i θ ) ) = e i m θ + e i ( m − 2 ) θ + ⋯ + e − i m θ . {\displaystyle \mathrm {X} \left({\begin{pmatrix}e^{i\theta }&0\\0&e^{-i\theta }\end{pmatrix}}\right)=e^{im\theta }+e^{i(m-2)\theta }+\cdots +e^{-im\theta }.} <p>The Weyl denominator, meanwhile, is simply the function e i θ − e − i θ {\displaystyle e^{i\theta }-e^{-i\theta }} . Multiplying the character by the Weyl denominator gives </p> X ( ( e i θ 0 0 e − i θ ) ) ( e i θ − e − i θ ) = ( e i ( m + 1 ) θ + e i ( m − 1 ) θ + ⋯ + e − i ( m − 1 ) θ ) − ( e i ( m − 1 ) θ + ⋯ + e − i ( m − 1 ) θ + e − i ( m + 1 ) θ ) . {\displaystyle \mathrm {X} \left({\begin{pmatrix}e^{i\theta }&0\\0&e^{-i\theta }\end{pmatrix}}\right)(e^{i\theta }-e^{-i\theta })=\left(e^{i(m+1)\theta }+e^{i(m-1)\theta }+\cdots +e^{-i(m-1)\theta }\right)-\left(e^{i(m-1)\theta }+\cdots +e^{-i(m-1)\theta }+e^{-i(m+1)\theta }\right).} <p>We can now easily verify that most of the terms cancel between the two term on the right-hand side above, leaving us with only </p> X ( ( e i θ 0 0 e − i θ ) ) ( e i θ − e − i θ ) = e i ( m + 1 ) θ − e − i ( m + 1 ) θ {\displaystyle \mathrm {X} \left({\begin{pmatrix}e^{i\theta }&0\\0&e^{-i\theta }\end{pmatrix}}\right)(e^{i\theta }-e^{-i\theta })=e^{i(m+1)\theta }-e^{-i(m+1)\theta }} <p>so that </p> X ( ( e i θ 0 0 e − i θ ) ) = e i ( m + 1 ) θ − e − i ( m + 1 ) θ e i θ − e − i θ = sin ( ( m + 1 ) θ ) sin θ . {\displaystyle \mathrm {X} \left({\begin{pmatrix}e^{i\theta }&0\\0&e^{-i\theta }\end{pmatrix}}\right)={\frac {e^{i(m+1)\theta }-e^{-i(m+1)\theta }}{e^{i\theta }-e^{-i\theta }}}={\frac {\sin((m+1)\theta )}{\sin \theta }}.} <p>The character in this case is a geometric series with R = e 2 i θ {\displaystyle R=e^{2i\theta }} and that preceding argument is a small variant of the standard derivation of the formula for the sum of a finite geometric series. </p> <h2 id="weyl-denominator-formula">Weyl denominator formula</h2> <p>In the special case of the trivial 1-dimensional representation the character is 1, so the Weyl character formula becomes the Weyl denominator formula:<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> </p> ∑ w ∈ W ε ( w ) e w ( ρ ) ( H ) = ∏ α ∈ Δ + ( e α ( H ) / 2 − e − α ( H ) / 2 ) . {\displaystyle {\sum _{w\in W}\varepsilon (w)e^{w(\rho )(H)}=\prod _{\alpha \in \Delta ^{+}}(e^{\alpha (H)/2}-e^{-\alpha (H)/2})}.} <p>For special unitary groups, this is equivalent to the expression </p> ∑ σ ∈ S n sgn ( σ ) X 1 σ ( 1 ) − 1 ⋯ X n σ ( n ) − 1 = ∏ 1 ≤ i < j ≤ n ( X j − X i ) {\displaystyle \sum _{\sigma \in S_{n}}\operatorname {sgn}(\sigma )\,X_{1}^{\sigma (1)-1}\cdots X_{n}^{\sigma (n)-1}=\prod _{1\leq i<j\leq n}(X_{j}-X_{i})} <p>for the <a href="/facts/Vandermonde_determinant/8oSbUk6u">Vandermonde determinant</a>.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> </p> <h2 id="weyl-dimension-formula">Weyl dimension formula</h2> <p>By evaluating the character at H = 0 {\displaystyle H=0} , Weyl's character formula gives the Weyl dimension formula </p> dim ( V λ ) = ∏ α ∈ Δ + ( λ + ρ , α ) ∏ α ∈ Δ + ( ρ , α ) {\displaystyle \dim(V_{\lambda })={\prod _{\alpha \in \Delta ^{+}}(\lambda +\rho ,\alpha ) \over \prod _{\alpha \in \Delta ^{+}}(\rho ,\alpha )}} <p>for the dimension of a finite dimensional representation V λ {\displaystyle V_{\lambda }} with highest weight λ {\displaystyle \lambda } . (As usual, ρ is half the sum of the positive roots and the products run over positive roots α.) The specialization is not completely trivial, because both the numerator and denominator of the Weyl character formula vanish to high order at the identity element, so it is necessary to take a limit of the trace of an element tending to the identity, using a version of <a href="/facts/L%2527H%25C3%25B4pital%2527s_rule/pCXK75Io">L'Hôpital's rule</a>.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> In the SU(2) case described above, for example, we can recover the dimension m + 1 {\displaystyle m+1} of the representation by using L'Hôpital's rule to evaluate the limit as θ {\displaystyle \theta } tends to zero of sin ( ( m + 1 ) θ ) / sin θ {\displaystyle \sin((m+1)\theta )/\sin \theta } . </p><p>We may consider as an example the complex semisimple Lie algebra sl(3,C), or equivalently the compact group SU(3). In that case, the <a href="/facts/Lie_algebra_representation/aaxnMxCR">representations</a> are labeled by a pair ( m 1 , m 2 ) {\displaystyle (m_{1},m_{2})} of non-negative integers. In this case, there are three positive roots and it is not hard to verify that the dimension formula takes the explicit form<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> </p> dim ( V m 1 , m 2 ) = 1 2 ( m 1 + 1 ) ( m 2 + 1 ) ( m 1 + m 2 + 2 ) {\displaystyle \dim(V_{m_{1},m_{2}})={\frac {1}{2}}(m_{1}+1)(m_{2}+1)(m_{1}+m_{2}+2)} <p>The case m 1 = 1 , m 2 = 0 {\displaystyle m_{1}=1,\,m_{2}=0} is the standard representation and indeed the dimension formula gives the value 3 in this case. </p> <h2 id="kostant-multiplicity-formula">Kostant multiplicity formula</h2> <p class="note">Main article: <a href="/facts/Kostant_partition_function/V7Kl9507">Kostant partition function</a></p> <p>The Weyl character formula gives the character of each representation as a quotient, where the numerator and denominator are each a finite linear combination of exponentials. While this formula in principle determines the character, it is not especially obvious how one can compute this quotient explicitly as a finite sum of exponentials. Already In the SU(2) case described above, it is not immediately obvious how to go from the Weyl character formula, which gives the character as sin ( ( m + 1 ) θ ) / sin θ {\displaystyle \sin((m+1)\theta )/\sin \theta } back to the formula for the character as a sum of exponentials: </p> e i m θ + e i ( m − 2 ) θ + ⋯ + e − i m θ . {\displaystyle e^{im\theta }+e^{i(m-2)\theta }+\cdots +e^{-im\theta }.} <p>In this case, it is perhaps not terribly difficult to recognize the expression sin ( ( m + 1 ) θ ) / sin θ {\displaystyle \sin((m+1)\theta )/\sin \theta } as the sum of a finite geometric series, but in general we need a more systematic procedure. </p><p>In general, the division process can be accomplished by computing a formal reciprocal of the Weyl denominator and then multiplying the numerator in the Weyl character formula by this formal reciprocal.<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> The result gives the character as a finite sum of exponentials. The coefficients of this expansion are the dimensions of the weight spaces, that is, the multiplicities of the weights. We thus obtain from the Weyl character formula a formula for the multiplicities of the weights, known as the Kostant multiplicity formula. An alternative formula, that is more computationally tractable in some cases, is given in the next section. </p> <h2 id="freudenthals-formula">Freudenthal's formula</h2> <p><a href="/facts/Hans_Freudenthal/l8V4uA0N">Hans Freudenthal</a>'s formula is a recursive formula for the weight multiplicities that gives the same answer as the Kostant multiplicity formula, but is sometimes easier to use for calculations as there can be far fewer terms to sum. The formula is based on use of the <a href="/facts/Casimir_element/rG71qcFb">Casimir element</a> and its derivation is independent of the character formula. It states<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> </p> ( ‖ Λ + ρ ‖ 2 − ‖ λ + ρ ‖ 2 ) m Λ ( λ ) = 2 ∑ α ∈ Δ + ∑ j ≥ 1 ( λ + j α , α ) m Λ ( λ + j α ) {\displaystyle (\|\Lambda +\rho \|^{2}-\|\lambda +\rho \|^{2})m_{\Lambda }(\lambda )=2\sum _{\alpha \in \Delta ^{+}}\sum _{j\geq 1}(\lambda +j\alpha ,\alpha )m_{\Lambda }(\lambda +j\alpha )} <p>where </p> <ul><li>Λ is a highest weight,</li> <li>λ is some other weight,</li> <li>mΛ(λ) is the multiplicity of the weight λ in the irreducible representation VΛ</li> <li>ρ is the Weyl vector</li> <li>The first sum is over all positive roots α.</li></ul> <h2 id="weylkac-character-formula">Weyl–Kac character formula</h2> <p>The Weyl character formula also holds for integrable <a href="/facts/Highest-weight_representation/7N8q5sw0">highest-weight representations</a> of <a href="/facts/Kac%25E2%2580%2593Moody_algebra/N8caXyvw">Kac–Moody algebras</a>, when it is known as the Weyl–Kac character formula. Similarly there is a denominator identity for Kac–Moody algebras, which in the case of the affine Lie algebras is equivalent to the <a href="/facts/Macdonald_identities/d2DtIaYG">Macdonald identities</a>. In the simplest case of the affine Lie algebra of type <i>A</i>1 this is the <a href="/facts/Jacobi_triple_product/VXIYGemW">Jacobi triple product</a> identity </p> ∏ m = 1 ∞ ( 1 − x 2 m ) ( 1 − x 2 m − 1 y ) ( 1 − x 2 m − 1 y − 1 ) = ∑ n = − ∞ ∞ ( − 1 ) n x n 2 y n . {\displaystyle \prod _{m=1}^{\infty }\left(1-x^{2m}\right)\left(1-x^{2m-1}y\right)\left(1-x^{2m-1}y^{-1}\right)=\sum _{n=-\infty }^{\infty }(-1)^{n}x^{n^{2}}y^{n}.} <p>The character formula can also be extended to integrable highest weight representations of <a href="/facts/Generalized_Kac%25E2%2580%2593Moody_algebra/gtlQfafQ">generalized Kac–Moody algebras</a>, when the character is given by </p> ∑ w ∈ W ( − 1 ) ℓ ( w ) w ( e λ + ρ S ) e ρ ∏ α ∈ Δ + ( 1 − e − α ) . {\displaystyle {\sum _{w\in W}(-1)^{\ell (w)}w(e^{\lambda +\rho }S) \over e^{\rho }\prod _{\alpha \in \Delta ^{+}}(1-e^{-\alpha })}.} <p>Here <i>S</i> is a correction term given in terms of the imaginary simple roots by </p> S = ∑ I ( − 1 ) | I | e Σ I {\displaystyle S=\sum _{I}(-1)^{|I|}e^{\Sigma I}\,} <p>where the sum runs over all finite subsets <i>I</i> of the imaginary simple roots which are pairwise orthogonal and orthogonal to the highest weight λ, and |I| is the cardinality of I and Σ<i>I</i> is the sum of the elements of <i>I</i>. </p><p>The denominator formula for the <a href="/facts/Monster_Lie_algebra/Kof19bN1">monster Lie algebra</a> is the product formula </p> j ( p ) − j ( q ) = ( 1 p − 1 q ) ∏ n , m = 1 ∞ ( 1 − p n q m ) c n m {\displaystyle j(p)-j(q)=\left({1 \over p}-{1 \over q}\right)\prod _{n,m=1}^{\infty }(1-p^{n}q^{m})^{c_{nm}}} <p>for the <a href="/facts/Elliptic_modular_function/H1a38AJF">elliptic modular function</a> <i>j</i>. </p><p>Peterson gave a recursion formula for the multiplicities mult(β) of the roots β of a symmetrizable (generalized) Kac–Moody algebra, which is equivalent to the Weyl–Kac denominator formula, but easier to use for calculations: </p> ( β , β − 2 ρ ) c β = ∑ γ + δ = β ( γ , δ ) c γ c δ {\displaystyle (\beta ,\beta -2\rho )c_{\beta }=\sum _{\gamma +\delta =\beta }(\gamma ,\delta )c_{\gamma }c_{\delta }\,} <p>where the sum is over positive roots γ, δ, and </p> c β = ∑ n ≥ 1 mult ( β / n ) n . {\displaystyle c_{\beta }=\sum _{n\geq 1}{\operatorname {mult} (\beta /n) \over n}.} <h2 id="harish-chandra-character-formula">Harish-Chandra Character Formula</h2> <p>Harish-Chandra showed that Weyl's character formula admits a generalization to representations of a real, <a href="/facts/Reductive_group/z1zyJvUh">reductive group</a>. Suppose π {\displaystyle \pi } is an irreducible, <a href="/facts/Admissible_representation/KjgWLM3s">admissible representation</a> of a real, reductive group G with <a href="/facts/Infinitesimal_character/wz43STaH">infinitesimal character</a> λ {\displaystyle \lambda } . Let Θ π {\displaystyle \Theta _{\pi }} be the <a href="/facts/Harish-Chandra_character/HycK1rxD">Harish-Chandra character</a> of π {\displaystyle \pi } ; it is given by integration against an <a href="/facts/Analytic_function/JhL3z8FP">analytic function</a> on the regular set. If H is a <a href="/facts/Cartan_subgroup/4mDR1t5Y">Cartan subgroup</a> of G and H' is the set of regular elements in H, then </p> Θ π | H ′ = ∑ w ∈ W / W λ a w e w λ e ρ ∏ α ∈ Δ + ( 1 − e − α ) . {\displaystyle \Theta _{\pi }|_{H'}={\sum _{w\in W/W_{\lambda }}a_{w}e^{w\lambda } \over e^{\rho }\prod _{\alpha \in \Delta ^{+}}(1-e^{-\alpha })}.} <p>Here </p> <ul><li>W is the complex Weyl group of H C {\displaystyle H_{\mathbb {C} }} with respect to G C {\displaystyle G_{\mathbb {C} }} </li> <li> W λ {\displaystyle W_{\lambda }} is the stabilizer of λ {\displaystyle \lambda } in W</li></ul> <p>and the rest of the notation is as above. </p><p>The coefficients a w {\displaystyle a_{w}} are still not well understood. Results on these coefficients may be found in papers of <a href="/facts/Rebecca_A._Herb/g14D3Fkm">Herb</a>, Adams, Schmid, and Schmid-Vilonen among others. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Character_theory/WMp5Q6YE">Character theory</a></li> <li><a href="/facts/Algebraic_character/AfMFyrzT">Algebraic character</a></li> <li><a href="/facts/Demazure_character_formula/1guQIDU8">Demazure character formula</a></li> <li><a href="/facts/Weyl_integration_formula/Wkfs0vn1">Weyl integration formula</a></li> <li><a href="/facts/Kirillov_character_formula/xXsrXuRG">Kirillov character formula</a></li></ul> <ul><li>Fulton, William and Harris, Joe (1991). <i>Representation theory: a first course.</i> New York: Springer-Verlag. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0387974954. OCLC 22861245.<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a></li> <li>Hall, Brian C. (2015), <i>Lie groups, Lie algebras, and representations: An elementary introduction</i>, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3319134666</li> <li>Humphreys, James E. (1972), <a href="https://archive.org/details/introductiontoli00jame"><i>Introduction to Lie Algebras and Representation Theory</i></a>, Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-90053-7.</li> <li><i>Infinite dimensional Lie algebras</i>, V. G. Kac, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-37215-1</li> <li>Duncan J. Melville (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Weyl–Kac_character_formula">"Weyl–Kac character formula"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a></li> <li><a href="/facts/Hermann_Weyl/4C5HhyGU">Weyl, Hermann</a> (1925), "Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen. I", <i><a href="/facts/Mathematische_Zeitschrift/8JqT7561">Mathematische Zeitschrift</a></i>, 23, Springer Berlin / Heidelberg: 271–309, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01506234">10.1007/BF01506234</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0025-5874">0025-5874</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:123145812">123145812</a></li> <li><a href="/facts/Hermann_Weyl/4C5HhyGU">Weyl, Hermann</a> (1926a), "Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen. II", <i><a href="/facts/Mathematische_Zeitschrift/8JqT7561">Mathematische Zeitschrift</a></i>, 24, Springer Berlin / Heidelberg: 328–376, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01216788">10.1007/BF01216788</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0025-5874">0025-5874</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:186229448">186229448</a></li> <li><a href="/facts/Hermann_Weyl/4C5HhyGU">Weyl, Hermann</a> (1926b), "Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen. III", <i><a href="/facts/Mathematische_Zeitschrift/8JqT7561">Mathematische Zeitschrift</a></i>, 24, Springer Berlin / Heidelberg: 377–395, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01216789">10.1007/BF01216789</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0025-5874">0025-5874</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:186232780">186232780</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Hall 2015 Section 12.4. - Hall, Brian C. (2015), Lie groups, Lie algebras, and representations: An elementary introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, ISBN 978-3319134666 <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Hall 2015 Section 10.4. - Hall, Brian C. (2015), Lie groups, Lie algebras, and representations: An elementary introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, ISBN 978-3319134666 <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Hall 2015 Section 12.5. - Hall, Brian C. (2015), Lie groups, Lie algebras, and representations: An elementary introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, ISBN 978-3319134666 <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Hall 2015 Theorem 10.14 - Hall, Brian C. (2015), Lie groups, Lie algebras, and representations: An elementary introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, ISBN 978-3319134666 <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Hall 2015 Section 10.4. - Hall, Brian C. (2015), Lie groups, Lie algebras, and representations: An elementary introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, ISBN 978-3319134666 <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Hall 2015 Section 12.3 - Hall, Brian C. (2015), Lie groups, Lie algebras, and representations: An elementary introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, ISBN 978-3319134666 <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>See Hall 2015 Section 10.8 in the Lie algebra setting and Section 12.4 in the compact group setting - Hall, Brian C. (2015), Lie groups, Lie algebras, and representations: An elementary introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, ISBN 978-3319134666 <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Hall 2015 Example 12.23 - Hall, Brian C. (2015), Lie groups, Lie algebras, and representations: An elementary introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, ISBN 978-3319134666 <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Hall 2015 Lemma 10.28. - Hall, Brian C. (2015), Lie groups, Lie algebras, and representations: An elementary introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, ISBN 978-3319134666 <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Hall 2015 Exercise 9 in Chapter 10. - Hall, Brian C. (2015), Lie groups, Lie algebras, and representations: An elementary introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, ISBN 978-3319134666 <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Hall 2015 Section 10.5. - Hall, Brian C. (2015), Lie groups, Lie algebras, and representations: An elementary introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, ISBN 978-3319134666 <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Hall 2015 Example 10.23 - Hall, Brian C. (2015), Lie groups, Lie algebras, and representations: An elementary introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, ISBN 978-3319134666 <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Hall 2015 Section 10.6 - Hall, Brian C. (2015), Lie groups, Lie algebras, and representations: An elementary introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, ISBN 978-3319134666 <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Humphreys 1972 Section 22.3 - Humphreys, James E. (1972), Introduction to Lie Algebras and Representation Theory, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90053-7 <a href="https://archive.org/details/introductiontoli00jame" target="_blank">https://archive.org/details/introductiontoli00jame</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Fulton, William, 1939- (1991). Representation theory : a first course. Harris, Joe, 1951-. New York: Springer-Verlag. ISBN 0387974954. OCLC 22861245.{{cite book}}: CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link) <a href="0387974954" target="_blank">0387974954</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> </ol>