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{
"Title": "Witten zeta values of simple Lie algebras",
"Definition": "For a complex simple Lie algebra $\\mathfrak g$ and a positive integer $m$, the Witten zeta value is $Z_{\\mathfrak g}(2m)=\\sum_{[V]\\in\\operatorname{Irr}_{\\mathrm{fd}}(\\mathfrak g)}(\\dim_{\\mathbb C}V)^{-2m}$, where the sum is over isomorphism classes of finite-dimensional irreducible complex representations CITE{Zhao}.",
"Parameters": {
"algebra": {
"type": "Symbolic",
"title": "simple Lie algebra",
"display": "$\\mathfrak g$",
"constraints": "a Cartan type of a complex simple Lie algebra, up to isomorphism",
"values": {
"A1": "$A_1$",
"A2": "$A_2$",
"B2": "$B_2=C_2$",
"G2": "$G_2$"
}
},
"m": {
"type": "Z",
"title": "half the zeta argument",
"display": "$m$",
"constraints": "$m\\geq1$"
}
},
"Comments": {
"representations": "The trivial representation contributes once. Distinct irreducible representations of the same dimension contribute separately.",
"global-form": "For compact groups, the convention is the simply connected form. In particular, $B_2=C_2$ denotes $\\mathfrak{so}_5(\\mathbb C)\\simeq\\mathfrak{sp}_4(\\mathbb C)$, or $\\operatorname{Spin}(5)\\simeq\\operatorname{Sp}(2)$, and occurs only once.",
"rank-one-explanation": "For $A_1=\\mathfrak{sl}_2(\\mathbb C)$, one irreducible representation has each positive integer dimension, so $Z_{A_1}(2m)$ is the HREF{Values_of_the_Riemann_zeta_function_at_rational_numbers}[Riemann zeta value] $\\zeta(2m)$. The compact forms of $A_1$ and $A_2$ are $SU(2)$ and $SU(3)$.",
"dimension-sums": "The positive integer variables $a,b$ in Formulas CITE{a2-sum}, CITE{b2-sum} and CITE{g2-sum} are shifted highest-weight coordinates. The corresponding representation dimensions are $ab(a+b)/2$, $ab(a+b)(a+2b)/6$ and $ab(a+b)(a+2b)(a+3b)(2a+3b)/120$ CITE{Zhao} CITE{KMT}. For $G_2$, individual HREF{Dimensions_of_the_irreducible_representations_of_the_exceptional_simple_Lie_algebras}[representation dimensions] are also tabulated.",
"normalization": "The root-system zeta functions of CITE{KMT} omit the factors $2^{2m}$, $6^{2m}$ and $120^{2m}$ in the three rank-two sums. Those factors are included here, so the trivial representation contributes $1$ in every case.",
"volume-formula": "The positive even special values occur in Witten volume formulas for moduli spaces of representations of surface fundamental groups CITE{Zhao} CITE{KMT}. The entries are zeta values; a moduli-space volume also depends on its geometric normalization.",
"classification": "The types $A_1,A_2,B_2=C_2,G_2$ exhaust complex simple Lie algebras of rank at most two. The reducible rank-two system $A_1\\times A_1$ is excluded CITE{RankTwo}."
},
"Tags": [
"special values",
"groups",
"L-function"
],
"Links": {
"RankTwo": {
"title": "MIT 18.745: Root systems, Lecture 21, Section 21.3",
"url": "https://ocw.mit.edu/courses/18-745-lie-groups-and-lie-algebras-i-fall-2020/mit18_745_f20_lec21.pdf"
}
},
"References": {
"Zhao": {
"bib": "J. Zhao, Witten volume formulas for semi-simple Lie algebras, preprint (2010), introduction, equation (1.1) and equation (2.1).",
"arxiv": "1001.3630"
},
"KMT": {
"bib": "Y. Komori, K. Matsumoto and H. Tsumura, On Witten multiple zeta-functions associated with semisimple Lie algebras IV, Glasgow Mathematical Journal 53 (2011), 185-206, equations (1.2)-(1.4), (2.4), and Example 2.2.",
"arxiv": "0907.0972",
"doi": "10.1017/S0017089510000613"
},
"ZhaoB2": {
"bib": "J. Zhao, Alternating Euler sums and special values of Witten multiple zeta function attached to so(5), MPIM preprint 2009-42 (2009), Section 4, page 8.",
"arxiv": "0903.0473"
}
},
"Data properties": {
"type": "R",
"digits": "100",
"rigour": "proven",
"complete": "no",
"complete-note": "it holds $A_1,A_2,B_2=C_2,G_2$ at every $m=1,2,3,4$, covering all simple complex Lie algebras of rank at most two up to isomorphism; the rank bound gives breadth across simply laced and non-simply-laced types, while the moment bound provides a low-order prefix of the even special-value family with independent exact examples for the harder rank-two types; both cutoffs are practical reference bounds",
"rigour details": "Exact rational partial fractions of the Weyl-chamber lattice sum give the coefficient $c_{\\mathfrak g,m}$ in Formula CITE{rational-pi-power}. Euler’s even-zeta formula is evaluated in exact rational arithmetic, and the final power of $\\pi$ is enclosed with Arb ball arithmetic at 429 working bits. The smallest measured ball accuracy is 423 bits, exceeding the 100 significant digits written. The attached generate.py and derivation.md give the algorithm and its justification; numerical-evidence.json records the exact coefficients and enclosure checks.\n\nAll 16 coefficients agree exactly with independent formulas: the first four even-zeta values for $A_1$, equation (1.1) of CITE{Zhao} for $A_2$, the four examples in Section 4 of CITE{ZhaoB2} for $B_2$, and Example 2.2 of CITE{KMT} for $G_2$. Separate 600-bit evaluations check the stored decimal intervals. Positive representation sums over $1\\leq a,b\\leq256$, with explicit integral tail bounds, check all 12 rank-two values. Known-answer and perturbation controls precede these comparisons. The attached checks.py reproduces these checks."
},
"Display properties": {
"number-header": "$Z_{\\mathfrak g}(2m)$"
},
"Numbers": {
"A1": {
"1": "1.644934066848226436472415166646025189218949901206798437735558229370007470403200873833628900619758705",
"2": "1.082323233711138191516003696541167902774750951918726907682976215444120616186968846556909635941699917",
"3": "1.017343061984449139714517929790920527901817490032853561842408664004332182901957897882773977938535171",
"4": "1.004077356197944339378685238508652465258960790649850020329110202652582952574748814395287230372371971"
},
"A2": {
"1": "1.356457415979265519619357239721227370535756653377138082456544885339109577202610530510365303918046894",
"2": "1.026784212342228425846358736056724563339973678841597320234592931085171393093666311128009212059066420",
"3": "1.002792560547090033501768052904745922465244530684347635688957098932436387704990396902423566700171169",
"4": "1.000306103576229548823689558291537684843430374044879333727890155547091799931400447664847037453066009"
},
"B2": {
"1": "1.129587025722687381801020893322234023416330889481081272870248977984155821646592416194698134168918468",
"2": "1.005657129606272367050681297721233511312368350552287071568530432524571904969524666578555135016047447",
"3": "1.000309351465608651994469835472343044161966054684543751884646138962512638084203312035447530587850736",
"4": "1.000017829741151859756552922974840278984888910937478370878366952718880726954254440695618072852287922"
},
"G2": {
"1": "1.027570527180418659255180091256990782373442051890000102922046323054117152305865582182200809200019478",
"2": "1.000444524236433370841971102399069431930469768301814496344892500080701813473674902487566490510283849",
"3": "1.000008635275437737441364241723920753920318072966575363363174978753476699200947081416692058343346547",
"4": "1.000000174147675051735167408976247014583001881393809638234901112042459382977932050990444247535626187"
}
},
"Formulas": {
"rank-one": "$Z_{A_1}(2m)=\\zeta(2m)$, with $\\zeta$ the HREF{Values_of_the_Riemann_zeta_function_at_rational_numbers}[Riemann zeta function].",
"a2-sum": "$\\displaystyle Z_{A_2}(2m)=2^{2m}\\sum_{a,b\\geq1}\\frac{1}{[ab(a+b)]^{2m}}$ CITE{Zhao}.",
"b2-sum": "$\\displaystyle Z_{B_2}(2m)=6^{2m}\\sum_{a,b\\geq1}\\frac{1}{[ab(a+b)(a+2b)]^{2m}}$ CITE{Zhao}.",
"g2-sum": "$\\displaystyle Z_{G_2}(2m)=120^{2m}\\sum_{a,b\\geq1}\\frac{1}{[ab(a+b)(a+2b)(a+3b)(2a+3b)]^{2m}}$ CITE{KMT}.",
"rational-pi-power": "$Z_{\\mathfrak g}(2m)=c_{\\mathfrak g,m}\\pi^{2mN}$ with $c_{\\mathfrak g,m}\\in\\mathbb Q_{>0}$, where $N$ is the number of positive roots. For $A_1,A_2,B_2,G_2$, the values of $N$ are $1,3,4,6$, respectively CITE{Zhao} CITE{KMT}.",
"a2-zeta-products": "$\\displaystyle Z_{A_2}(2m)=\\frac{4^{m+1}}{3}\\sum_{\\substack{0\\leq i\\leq2m\\\\i\\;\\mathrm{even}}}\\binom{4m-i-1}{2m-1}\\zeta(i)\\zeta(6m-i)$, with $\\zeta(0)=-1/2$ CITE{Zhao}.",
"first-values": "$\\displaystyle Z_{A_1}(2)=\\frac{\\pi^2}{6},\\quad Z_{A_2}(2)=\\frac{4\\pi^6}{2835},\\quad Z_{B_2}(2)=\\frac{\\pi^8}{8400},\\quad Z_{G_2}(2)=\\frac{460\\pi^{12}}{413756343}$ CITE{Zhao} CITE{ZhaoB2} CITE{KMT}."
},
"Similar tables": [
{
"table": "HREF{Values_of_the_Riemann_zeta_function_at_rational_numbers}[Values of the Riemann zeta function]",
"relation": "coincide with the $A_1$ values at the positive even arguments"
},
{
"table": "HREF{Dimensions_of_the_irreducible_representations_of_the_exceptional_simple_Lie_algebras}[Dimensions of exceptional Lie algebra representations]",
"relation": "provide individual dimensions that enter the $G_2$ representation sum"
}
],
"Keywords": [
"representation zeta function",
"Witten volume formula",
"SU(2)",
"SU(3)",
"Spin(5)",
"Sp(2)",
"B2",
"C2",
"G2",
"Mordell-Tornheim",
"Witten zeta function"
],
"Programs": {
"Sage": {
"code": "# Install the numberdb client in SageMath once: sage -pip install numberdb\n# Download this table's attached generate.py into the working directory.\n# Run this code in SageMath from that directory.\nfrom generate import WittenZeta\ng = WittenZeta()\n# Change algebra and m to request another positive even argument s = 2*m.\nprint(g.value({'algebra': 'G2', 'm': '5'}, 100))",
"language": "Sage"
}
}
}