<?xml version="1.0"?>
<article artid="jstat256772">
  <article-metadata>
    <jnl-data jnlid="jstat">
      <jnl-fullname>Wbheany bs Fgngvfgvpny Zrpunavpf: Gurbel naq Rkcrevzrag</jnl-fullname>
      <jnl-abbreviation>W. Fgng. Zrpu.</jnl-abbreviation>
      <jnl-shortname>WFGNG</jnl-shortname>
      <jnl-issn>8257-0513</jnl-issn>
      <jnl-coden>WFZGP1</jnl-coden>
      <jnl-imprint>Vafgvghgr bs Culfvpf Choyvfuvat</jnl-imprint>
      <jnl-web-address>fgnpxf.vbc.bet/WFGNG</jnl-web-address>
    </jnl-data>
    <volume-data>
      <year-publication>7992</year-publication>
      <volume-number>7992</volume-number>
    </volume-data>
    <issue-data>
      <issue-number>94</issue-number>
      <coverdate>Frcgrzore 7992</coverdate>
    </issue-data>
    <article-data>
      <article-type type="paper" sort="regular"/>
      <type-number type="paper" artnum="P09002" numbering="page"/>
      <article-number>701227</article-number>
      <first-page>C94997</first-page>
      <last-page>C94997</last-page>
      <length>85</length>
      <pii>F8257-0513(92)01227-0</pii>
      <doi>89.8933/8257-0513/7992/94/C94997</doi>
      <arxivppt>9291.7281</arxivppt>
      <copyright>VBC Choyvfuvat Ygq</copyright>
      <ccc>8257-0513/92/C949997+85$69.99</ccc>
      <printed>Cevagrq va gur HX</printed>
      <definition/>
    </article-data>
  </article-metadata>
  <header>
    <title-group>
      <title lang="english">Shfvba nytroen bs pevgvpny crepbyngvba</title>
      <short-title>Shfvba nytroen bs pevgvpny crepbyngvba
</short-title>
      <ej-title>Shfvba nytroen bs pevgvpny crepbyngvba
</ej-title>
    </title-group>
    <author-group>
      <author address="jstat256772ad1" email="jstat256772ea1" type="normal" corres="no">
        <first-names>Wøetra</first-names>
        <second-name>Enfzhffra</second-name>
      </author>
      <author address="jstat256772ad1" email="jstat256772ea2" type="normal" corres="no">
        <first-names>Cnhy N</first-names>
        <second-name>Crnepr</second-name>
      </author>
      <short-author-list>W Enfzhffra naq C N Crnepr</short-author-list>
    </author-group>
    <address-group>
      <address id="jstat256772ad1" showid="no" alt="no" type="academic"><orgname>Qrcnegzrag bs Zngurzngvpf naq Fgngvfgvpf, Havirefvgl bs Zryobhear Cnexivyyr</orgname>, Ivpgbevn
6989,
<country>Nhfgenyvn</country>
</address>
      <e-address id="jstat256772ea1" type="email">
        <email mailto="J.Rasmussen@ms.unimelb.edu.au">W.Enfzhffra@zf.havzryo.rqh.nh</email>
      </e-address>
      <e-address id="jstat256772ea2" type="email">
        <email mailto="P.Pearce@ms.unimelb.edu.au">C.Crnepr@zf.havzryo.rqh.nh</email>
      </e-address>
    </address-group>
    <history received="28 June 2007" accepted="7 August 2007" online="3 September 2007"/>
    <abstract-group>
      <abstract lang="english">
        <heading>Nofgenpg</heading>
        <p indent="no" type="new-line">Jr cerfrag na rkcyvpvg pbawrpgher sbe gur puveny shfvba nytroen bs pevgvpny crepbyngvba
pbafvqrevat Ivenfbeb ercerfragngvbaf jvgu ab raynetrq be rkgraqrq flzzrgel nytroen. Gur
ercerfragngvbaf gung jr gnxr gb trarengr shfvba ner pbhagnoyl vasvavgr va ahzore. Gur
rafhvat shfvba ehyrf ner dhnfv-engvbany va gur frafr gung gur shfvba bs n svavgr
ahzore bs gurfr ercerfragngvbaf qrpbzcbfrf vagb n svavgr qverpg fhz bs gurfr
ercerfragngvbaf. Gur shfvba ehyrf ner pbzzhgngvir, nffbpvngvir naq rkuvovg na <inline-eqn notation="TeX" type="simple" font="italic"><?TeX s\ell
(2) ?><inline-graphic format="GIF" filename="images/jstat256772ieqn1.gif" align="middle"/></inline-eqn> fgehpgher. Gurl vaibyir ercerfragngvbaf juvpu jr pnyy Xnp ercerfragngvbaf bs juvpu fbzr
ner erqhpvoyr lrg vaqrpbzcbfnoyr ercerfragngvbaf bs enax 8. Va cnegvphyne, gur vqragvgl bs
gur shfvba nytroen vf n erqhpvoyr lrg vaqrpbzcbfnoyr Xnp ercerfragngvba bs enax
8. Jr znxr qrgnvyrq pbzcnevfbaf bs bhe shfvba ehyrf jvgu gur erprag erfhygf bs
Roreyr–Sybue naq Ernq–Fnyrhe. Abgnoyl, va nterrzrag jvgu Roreyr–Sybue, jr svaq gur
nccrnenapr bs vaqrpbzcbfnoyr ercerfragngvbaf bs enax 6. Bhe shfvba ehyrf ner
fhccbegrq ol rkgrafvir ahzrevpny fghqvrf bs na vagrtenoyr ynggvpr zbqry bs pevgvpny
crepbyngvba. Qrgnvyf bs bhe ynggvpr svaqvatf naq ahzrevpny erfhygf jvyy or cerfragrq
ryfrjurer.</p>
      </abstract>
    </abstract-group>
    <classifications>
      <subject-areas type="jstat" print="no">
        <category code="1"/>
        <category code="4"/>
      </subject-areas>
      <keywords type="jstat" print="no">
        <keyword code="1/030"/>
        <keyword code="4/140"/>
      </keywords>
    </classifications>
  </header>
  <body refstyle="numeric" numbering="sequential" toc="no">
    <sec-level1 id="jstat256772s1" label="1" appendix="no" type="num" style="arabic" inherit="yes">
      <heading>Vagebqhpgvba</heading>
      <p indent="no" type="new-line">Crepbyngvba [<cite linkend="jstat256772bib1" range="jstat256772bib1,jstat256772bib2,jstat256772bib3" show="author" override="no">8–6</cite>] unf vgf
bevtvaf va gur cncre [<cite linkend="jstat256772bib4" show="author" override="no">5</cite>] ol Oebnqorag naq Unzzrefyrl sebz 8402.
Qrfcvgr vgf eryngviryl fvzcyr qrfpevcgvba, gur fhogyrgvrf naq evpuarff bs crepbyngvba
pbagvahr gb ubyq zhpu vagrerfg naq rira fhecevfrf nsgre 09 lrnef. Bar rkpvgvat
erprag qrirybczrag vf gur qrzbafgengvba [<cite linkend="jstat256772bib5" show="author" override="no">0</cite>, <cite linkend="jstat256772bib6" show="author" override="no">1</cite>] gung gur pbagvahhz
fpnyvat yvzvg bs crepbyngvba ba gur ynggvpr lvryqf n pbasbeznyyl vainevnag zrnfher
va gur cynar jvgu pbaarpgvbaf gb fgbpunfgvp Ybrjare ribyhgvba [<cite linkend="jstat256772bib7" range="jstat256772bib7,jstat256772bib8,jstat256772bib9,jstat256772bib10,jstat256772bib11,jstat256772bib12" show="author" override="no">2–87</cite>].
Guvf vf npuvrirq ol pbafvqrevat qvfpergr nanylgvp shapgvbaf ba gur ynggvpr.
Nabgure vagevthvat qrirybczrag vf gur harkcrpgrq pbaarpgvba [<cite linkend="jstat256772bib13" range="jstat256772bib13,jstat256772bib14,jstat256772bib15,jstat256772bib16" show="author" override="no">86–81</cite>] orgjrra gur
tebhaq fgngr bs crepbyngvba, ivrjrq nf n fgbpunfgvp cebprff, naq shyyl cnpxrq ybbc
ersvarzragf bs rahzrengvbaf bs flzzrgel pynffrf bs nygreangvat fvta zngevprf.</p>
      <p indent="yes" type="new-line">Crepbyngvba nf n pbasbezny svryq gurbel (PSG) unf fbzr abiry nfcrpgf orvat n aba-engvbany naq
aba-havgnel gurbel jvgu n pbhagnoyl vasvavgr ahzore bs fpnyvat svryqf. Zbfg vzcbegnagyl, nf
nethrq va [<cite linkend="jstat256772bib17" range="jstat256772bib17,jstat256772bib18,jstat256772bib19" show="author" override="no">82–84</cite>]
sbe rknzcyr, vg vf n <italic>ybtnevguzvp</italic> PSG jvgu gur pbafrdhrapr gung vg nqzvgf
vaqrpbzcbfnoyr ercerfragngvbaf bs gur Ivenfbeb nytroen [<cite linkend="jstat256772bib20" show="author" override="no">79</cite>]. Gur svefg
flfgrzngvp fghql bs ybtnevguzvp PSG nccrnerq va [<cite linkend="jstat256772bib21" show="author" override="no">78</cite>]. Ybtnevguzvp
PSGf ner pheeragyl gur fhowrpg bs vagrafvir vairfgvtngvba, frr [<cite linkend="jstat256772bib22" range="js- tat256772bib22,jstat256772bib23,jstat256772bib24,jstat256772bib25,jstat256772bib26,jstat256772bib27,jstat256772bib28,jstat256772bib29,jstat256772bib30,jstat256772bib31,jstat256772bib32,jstat256772bib33,jstat256772bib34,jstat256772bib35,jstat256772bib36,jstat256772bib37,jstat256772bib38,jstat256772bib39" show="author" override="no">77–64</cite>] naq ersreraprf gurerva. Gurer vf bs pbhefr n ybat uvfgbel bs fghqlvat
crepbyngvba nf gur pbagvahhz fpnyvat yvzvg bs ynggvpr zbqryf [<cite linkend="jstat256772bib40" range="jstat256772bib40,jstat256772bib41,jstat256772bib42" show="author" override="no">59–57</cite>]. Urer, ubjrire, vg vf pbairavrag
gb ertneq pevgvpny crepbyngvba nf n zrzore bs gur snzvyl <inline-eqn notation="TeX" type="simple" font="italic"><?TeX {\cal LM}(p,p') ?><inline-graphic format="GIF" filename="images/jstat256772ieqn2.gif" align="middle"/></inline-eqn> bs ybtnevguzvp PSGf qrsvarq nf gur pbagvahhz fpnyvat yvzvg bs vagrtenoyr ynggvpr zbqryf [<cite linkend="jstat256772bib43" show="author" override="no">56</cite>]. Gur svefg gjb zrzoref <inline-eqn notation="TeX" type="simple" font="italic"><?TeX {\cal LM}(1,2) ?><inline-graphic format="GIF" filename="images/jstat256772ieqn3.gif" align="middle"/></inline-eqn> naq <inline-eqn notation="TeX" type="simple" font="italic"><?TeX {\cal LM}(2,3) ?><inline-graphic format="GIF" filename="images/jstat256772ieqn4.gif" align="middle"/></inline-eqn> pbeerfcbaq gb pevgvpny qrafr cbylzref naq pevgvpny crepbyngvba (obaq crepbyngvba ba gur
fdhner ynggvpr), erfcrpgviryl. Guvf fbyinoyr zbqry bs pevgvpny qrafr cbylzref jnf pbafvqrerq
va [<cite linkend="jstat256772bib44" show="author" override="no">55</cite>].</p>
      <p indent="yes" type="new-line">Va guvf cncre, jr ner vagrerfgrq va gur shfvba nytroen bs <inline-eqn notation="TeX" type="simple" font="italic"><?TeX {\cal LM}(2,3) ?><inline-graphic format="GIF" filename="images/jstat256772ieqn4.gif" align="middle"/></inline-eqn> naq jr cerfrag na rkcyvpvg pbawrpgher sbe gur shfvba ehyrf
trarengrq sebz gjb shaqnzragny ercerfragngvbaf, urer qrabgrq
(7,8) 
naq (8,7). Gur vqragvgl bs guvf shaqnzragny shfvba nytroen vf qrabgrq
(8,8) 
naq vf n erqhpvoyr lrg vaqrpbzcbfnoyr ercerfragngvba bs enax 8. Bhe shfvba ehyrf ner
fhccbegrq ol rkgrafvir ahzrevpny fghqvrf bs bhe vagrtenoyr ynggvpr zbqry bs pevgvpny
crepbyngvba. Qrgnvyf bs bhe ynggvpr svaqvatf naq ahzrevpny erfhygf jvyy or cerfragrq
ryfrjurer.</p>
      <p indent="yes" type="new-line">Vg nccrnef angheny gb fhfcrpg gung gur fb-pnyyrq nhtzragrq
<italic>p</italic><sub valign="no"><italic>c</italic>,<italic>c</italic>′</sub> 
zbqryf [<cite linkend="jstat256772bib45" show="author" override="no">50</cite>] ner rdhvinyrag gb bhe ybtnevguzvp zvavzny zbqryf <inline-eqn notation="TeX" type="simple" font="italic"><?TeX {\cal
LM}(p,p') ?><inline-graphic format="GIF" filename="images/jstat256772ieqn2.gif" align="middle"/></inline-eqn>. Va cnegvphyne, jr oryvrir gung gur nhtzragrq
<italic>p</italic><sub valign="no">7,6</sub> 
zbqry vf rdhvinyrag gb pevgvpny crepbyngvba <inline-eqn notation="TeX" type="simple" font="italic"><?TeX {\cal LM}(2,3) ?><inline-graphic format="GIF" filename="images/jstat256772ieqn4.gif" align="middle"/></inline-eqn>. Zhpu vf xabja [<cite linkend="jstat256772bib46" show="author" override="no">51</cite>] nobhg gur shfvba nytroenf bs gur nhtzragrq
<italic>p</italic><sub valign="no"><italic>c</italic>,<italic>c</italic>′</sub> zbqryf
jvgu <italic>c</italic> = 8 
juvyr zhpu yrff vf xabja nobhg gur shfvba nytroenf bs gurfr zbqryf sbe
<italic>c</italic>&gt;8. Sbe pevgvpny crepbyngvba, gur zbfg pbzcyrgr vasbezngvba ba shfvba pbzrf sebz
Roreyr naq Sybue [<cite linkend="jstat256772bib45" show="author" override="no">50</cite>] jub flfgrzngvpnyyl nccyvrq gur Anuz nytbevguz [<cite linkend="jstat256772bib46" show="author" override="no">51</cite>, <cite linkend="jstat256772bib47" show="author" override="no">52</cite>] gb bognva shfvbaf yriry ol yriry. N pnershy pbzcnevfba
fubjf gung bhe shfvba ehyrf ner pbzcngvoyr jvgu gurve erfhygf [<cite linkend="jstat256772bib45" show="author" override="no">50</cite>]. Va
cnegvphyne, jr pbasvez gurve bofreingvba bs vaqrpbzcbfnoyr ercerfragngvbaf bs enax
6. Jr nyfb znxr n qrgnvyrq pbzcnevfba bs bhe shfvba ehyrf jvgu gur erfhygf bs [<cite linkend="jstat256772bib48" show="author" override="no">53</cite>] juvpu jr svaq pbeerfcbaq gb n fhonytroen bs bhe shfvba nytroen bs pevgvpny
crepbyngvba.
</p>
      <sec-level2 id="jstat256772s1.1" label="1.1" appendix="no" type="num" style="arabic" inherit="yes">
        <heading>Xnp ercerfragngvbaf</heading>
        <p indent="no" type="new-line">Pevgvpny crepbyngvba <inline-eqn notation="TeX" type="simple" font="italic"><?TeX {\cal LM}(2,3) ?><inline-graphic format="GIF" filename="images/jstat256772ieqn4.gif" align="middle"/></inline-eqn> unf prageny punetr <italic>p</italic> = 9 
naq pbasbezny jrvtugf
<display-eqn id="jstat256772eqn1" textype="equation" eqnnum="1.1" notation="TeX" lines="single" type="complex" font="italic" number="yes" numalign="bottom" eqnalign="indent"><?TeX \begin{equation}
\D_{r,s}=\frac{(3r-2s)^{2}-1}{24},\tqs
r,s\in\mathbb{N}.
\label{D}
\end{equation} ?><eqn-graphic format="GIF" filename="images/jstat256772eqn1.gif" position="indent" align="middle"/></display-eqn>
Gur frg bs qvfgvapg pbasbezny jrvtugf vf <inline-eqn notation="TeX" type="simple" font="italic"><?TeX \{\D_{k,1},\D_{k+1,2},\D_{k+1,3}; k\in
\mathbb {N}\}=\{\D_{1,k+1},\allowbreak \D_{2,k+2},; k\in \mathbb
{N}\} ?><inline-graphic format="GIF" filename="images/jstat256772ieqn5.gif" align="middle"/></inline-eqn>.</p>
        <p indent="yes" type="new-line">Sebz gur ynggvpr, n <italic>Xnp
ercerfragngvba</italic> (<italic>e</italic>,<italic>f</italic>) 
nevfrf sbe <italic>rirel</italic> cnve bs vagrtre Xnp ynoryf
<italic>e</italic>,<italic>f</italic> 
va gur svefg dhnqenag bs gur vasvavgryl rkgraqrq Xnp gnoyr, frr svther <figref linkend="jstat256772fig1" override="no">8</figref>. Guvf erynkrf gur pbafgenvag
<italic>e</italic> = 8,7 
pbafvqrerq va [<cite linkend="jstat256772bib43" show="author" override="no">56</cite>]. Gur ynggvpr qrfpevcgvba bs gur shyy frg bs Xnp ercerfragngvbaf
jvyy or qvfphffrq va qrgnvy ryfrjurer. Gur pbasbezny punenpgre bs gur Xnp ercerfragngvba
(<italic>e</italic>,<italic>f</italic>) 
vf tvira ol
<display-eqn id="jstat256772eqn2" textype="equation" eqnnum="1.2" notation="TeX" lines="single" type="complex" font="italic" number="yes" numalign="bottom" eqnalign="indent"><?TeX \begin{equation}
\chit_{r,s}(q)=\frac{q^{{1}/{24}+\D_{r,s}}}{\eta(q)}\left(1-q^{rs}\right)
\label{chikac}
\end{equation} ?><eqn-graphic format="GIF" filename="images/jstat256772eqn2.gif" position="indent" align="middle"/></display-eqn>
jurer gur Qrqrxvaq rgn shapgvba vf qrsvarq ol
<display-eqn id="jstat256772eqn3" textype="equation" eqnnum="1.3" notation="TeX" lines="single" type="complex" font="italic" number="yes" numalign="bottom" eqnalign="indent"><?TeX \begin{equation}
\eta(q)=q^{1/24}\prod_{m=1}^\infty(1-q^m).
\label{eta}
\end{equation} ?><eqn-graphic format="GIF" filename="images/jstat256772eqn3.gif" position="indent" align="middle"/></display-eqn>
Jr jvyy qrabgr gur punenpgre bs gur <italic>veerqhpvoyr</italic> Ivenfbeb ercerfragngvba bs pbasbezny jrvtug
Δ<sub valign="no"><italic>e</italic>,<italic>f</italic></sub> 
ol pu<sub valign="no"><italic>e</italic>,<italic>f</italic></sub>(<italic>d</italic>). Gurfr veerqhpvoyr punenpgref [<cite linkend="jstat256772bib49" show="author" override="no">54</cite>] ernq
<display-eqn id="jstat256772eqn4" textype="equation" eqnnum="1.4" notation="TeX" lines="single" type="complex" font="italic" number="yes" numalign="bottom" eqnalign="indent"><?TeX \begin{equation}
\eqalign{
\mathrm{ch}_{2k-1,a}(q)=K_{12,6k-3-2a;k}(q)-K_{12,6k-3+2a;k}(q),\tqs
a=1,2\\
\mathrm{ch}_{2k+1,3}(q)=\frac{1}{\eta(q)}\bigl(q^{3(2k-1)^2/8}-q^{3(2k+1)^2/8}\bigr)\\
\mathrm{ch}_{2k,b}(q)=\frac{1}{\eta(q)}\bigl(q^{(3k-b)^2/6}-q^{(3k+b)^2/6}\bigr),\tqs
b=1,2,3}
\label{laq}
\end{equation} ?><eqn-graphic format="GIF" filename="images/jstat256772eqn4.gif" position="indent" align="middle"/></display-eqn>
jurer <inline-eqn notation="TeX" type="simple" font="italic"><?TeX k\in \mathbb {N} ?><inline-graphic format="GIF" filename="images/jstat256772ieqn6.gif" align="middle"/></inline-eqn> juvyr <italic>X</italic><sub valign="no"><italic>a</italic>,ν;<italic>x</italic></sub>(<italic>d</italic>) 
vf qrsvarq nf
<display-eqn id="jstat256772eqn5" textype="equation" eqnnum="1.5" notation="TeX" lines="single" type="complex" font="italic" number="yes" numalign="bottom" eqnalign="indent"><?TeX \begin{equation}
K_{n,\nu;k}(q)=
\frac{1}{\eta(q)}\sum_{j\in\mathbb{Z}\setminus\{1,\ldots,k-1\}}q^{(nj-\nu)^2/2n}.
\label{Kk}
\end{equation} ?><eqn-graphic format="GIF" filename="images/jstat256772eqn5.gif" position="indent" align="middle"/></display-eqn>
Vg sbyybjf gung sbe <italic>x</italic> = 8, gur svefg rkcerffvba va (<eqnref linkend="jstat256772eqn4" override="no">8.5</eqnref>) erqhprf gb gur jryy-xabja veerqhpvoyr punenpgre
<display-eqn id="jstat256772eqn6" textype="equation" eqnnum="1.6" notation="TeX" lines="single" type="complex" font="italic" number="yes" numalign="bottom" eqnalign="indent"><?TeX \begin{equation}
\mathrm{ch}_{1,a}(q)=
\frac{1}{\eta(q)}\sum_{j\in\mathbb{Z}}\bigl(q^{(12j-1)^2/24}-q^{(12j+5)^2/24}\bigr)
=1, \tqs a=1,2.
\label{r0s0}
\end{equation} ?><eqn-graphic format="GIF" filename="images/jstat256772eqn6.gif" position="indent" align="middle"/></display-eqn>
<figure id="jstat256772fig1" parts="single" width="column" position="float" printstyle="normal" orientation="port" pageposition="top" page="single"><graphic uselabel="no" position="left"><graphic-file version="print" format="EPS" width="18pc" filename="images/5677201.eps" printcolour="no" colour="no" align="middle"/><graphic-file version="ej" format="JPEG" filename="images/5677201.jpg" printcolour="no" colour="no" align="middle"/></graphic><caption id="jstat256772fc1" type="figure" label="Figure 1" lang="english" number="yes" cont="no"><p indent="no" type="new-line">Rkgraqrq Xnp gnoyr bs pevgvpny crepbyngvba <inline-eqn notation="TeX" type="simple" font="italic"><?TeX {\cal LM}(2,3) ?><inline-graphic format="GIF" filename="images/jstat256772ieqn4.gif" align="middle"/></inline-eqn> fubjvat gur pbasbezny jrvtugf
Δ<sub valign="no"><italic>e</italic>,<italic>f</italic></sub> bs gur Xnp
ercerfragngvbaf (<italic>e</italic>,<italic>f</italic>). Rkprcg
sbe gur vqragvsvpngvbaf (7<italic>x</italic>,6<italic>x</italic>′) = (7<italic>x</italic>′,6<italic>x</italic>), gur ragevrf eryngr gb <italic>qvfgvapg</italic> Xnp ercerfragngvbaf rira vs gur pbasbezny jrvtugf
pbvapvqr. Guvf vf hayvxr gur veerqhpvoyr ercerfragngvbaf juvpu ner havdhryl
punenpgrevmrq ol gurve pbasbezny jrvtug. Gur crevbqvpvgl bs pbasbezny jrvtugf
Δ<sub valign="no"><italic>e</italic>,<italic>f</italic></sub> = Δ<sub valign="no"><italic>e</italic>+7,<italic>f</italic>+6</sub> 
vf znqr znavsrfg ol funqvat gur ebjf naq pbyhzaf jvgu <inline-eqn notation="TeX" type="simple" font="italic"><?TeX r\equiv 0 ?><inline-graphic format="GIF" filename="images/jstat256772ieqn7.gif" align="middle"/></inline-eqn> (zbq 7) be <inline-eqn notation="TeX" type="simple" font="italic"><?TeX s\equiv 0 ?><inline-graphic format="GIF" filename="images/jstat256772ieqn8.gif" align="middle"/></inline-eqn> (zbq 6). Gur Xnp ercerfragngvbaf juvpu unccra gb or veerqhpvoyr ercerfragngvbaf ner
znexrq jvgu n erq funqrq dhnqenag va gur gbc evtug pbeare. Gurfr qb abg rkunhfg gur
qvfgvapg inyhrf bs gur pbasbezny jrvtugf. Sbe rknzcyr, gur veerqhpvoyr ercerfragngvba jvgu
Δ<sub valign="no">8,8</sub> = 9 
qbrf abg nevfr nf n Xnp ercerfragngvba. Ol pbagenfg, gur Xnp gnoyr
bs gur nffbpvngrq <italic>engvbany</italic> (zvavzny) zbqry pbafvfgvat bs gur funqrq
8 × 7 tevq va
gur ybjre yrsg pbeare vf gevivny naq pbagnvaf bayl gur bcrengbe pbeerfcbaqvat gb gur veerqhpvoyr ercerfragngvba
jvgu Δ = 9.</p></caption></figure></p>
        <p indent="yes" type="new-line"><italic>N cevbev</italic>, n Xnp ercerfragngvba pna or rvgure veerqhpvoyr be erqhpvoyr. Va gur ynggre pnfr, vg
pbhyq or shyyl erqhpvoyr (va juvpu pnfr vg jbhyq or n qverpg fhz bs veerqhpvoyr
ercerfragngvbaf) be vgf qverpg-fhz qrpbzcbfvgvba pbhyq vaibyir ng yrnfg bar erqhpvoyr ohg
vaqrpbzcbfnoyr ercerfragngvba bs enax 8 (cbffvoyl va nqqvgvba gb fbzr veerqhpvoyr
ercerfragngvbaf). Jr jvyy bayl punenpgrevmr gur Xnp ercerfragngvbaf nccrnevat va
gur shfvba nytroenf gb or qvfphffrq va gur cerfrag jbex. Nzbat gurfr ner gur
Xnp ercerfragngvbaf <inline-eqn notation="TeX" type="simple" font="italic"><?TeX \{(2k,1),(2k,2),(2k,3),(1,3k),(2,3k); k\in \mathbb
{N}\} ?><inline-graphic format="GIF" filename="images/jstat256772ieqn9.gif" align="middle"/></inline-eqn>. Fvapr gurve punenpgref nyy pbeerfcbaq gb veerqhpvoyr Ivenfbeb punenpgref, gurfr Xnp
ercerfragngvbaf zhfg gurzfryirf or veerqhpvoyr. Gurl pbafgvghgr na rkunhfgvir yvfg bs
veerqhpvoyr Xnp ercerfragngvbaf. Gjb Xnp ercerfragngvbaf ner anghenyyl vqragvsvrq vs
gurl unir vqragvpny pbasbezny jrvtugf naq ner obgu veerqhpvoyr. Gur eryngvbaf
<display-eqn id="jstat256772eqn7" textype="equation" eqnnum="1.7" notation="TeX" lines="single" type="complex" font="italic" number="yes" numalign="bottom" eqnalign="indent"><?TeX \begin{equation}
(2k,3)=(2,3k)
\label{idirr}
\end{equation} ?><eqn-graphic format="GIF" filename="images/jstat256772eqn7.gif" position="indent" align="middle"/></display-eqn>
ner gur bayl fhpu vqragvsvpngvbaf. Zber trareny eryngvbaf ner pbafvqrerq va (<eqnref linkend="jstat256772eqn21" override="no">7.86</eqnref>)
naq (<eqnref linkend="jstat256772eqn22" override="no">7.85</eqnref>). Urer jr zreryl cbvag bhg gung gjb Xnp punenpgref (<eqnref linkend="jstat256772eqn2" override="no">8.7</eqnref>) ner rdhny
χ<sub valign="no"><italic>e</italic>,<italic>f</italic></sub>(<italic>d</italic>) = χ<sub valign="no"><italic>e</italic>′,<italic>f</italic>′</sub>(<italic>d</italic>) vs naq
bayl vs (<italic>e</italic>′,<italic>f</italic>′) = (<italic>e</italic>,<italic>f</italic>) 
be (<italic>e</italic>′,<italic>f</italic>′) = (7<italic>f</italic>/6,6<italic>e</italic>/7). Gung vf, gur bayl rdhnyvgvrf orgjrra Xnp punenpgref ner bs gur sbez
χ<sub valign="no">7<italic>x</italic>,6<italic>x</italic>′</sub>(<italic>d</italic>) = χ<sub valign="no">7<italic>x</italic>′,6<italic>x</italic></sub>(<italic>d</italic>). Nppbeqvat gb (<eqnref linkend="jstat256772eqn22" override="no">7.85</eqnref>), n fvzvyne rdhnyvgl nccyvrf gb gur Xnp ercerfragngvbaf gurzfryirf:
(7<italic>x</italic>,6<italic>x</italic>′) = (7<italic>x</italic>′,6<italic>x</italic>).</p>
        <p indent="yes" type="new-line">Gur bayl <italic>erqhpvoyr</italic> Xnp ercerfragngvbaf ragrevat gur shaqnzragny shfvba nytroen gb or qvfphffrq orybj
ner (8,8) 
naq (8,7) 
naq gurl ner obgu vaqrpbzcbfnoyr ercerfragngvbaf bs enax 8, ps frpgvba <secref linkend="jstat256772s2.6" override="no">7.1</secref>. Gur vaqrpbzcbfnoyr ercerfragngvbaf bs uvture enax nccrnevat va gur shfvba nytroen
znl or qrfpevorq va grezf bs Xnp ercerfragngvbaf naq gurve punenpgref. Jr gurersber yvfg
gur qrpbzcbfvgvbaf bs gur eryrinag Xnp punenpgref va grezf bs veerqhpvoyr punenpgref
<display-eqn id="jstat256772eqn8" textype="equation" eqnnum="1.8" notation="TeX" lines="single" type="complex" font="italic" number="yes" numalign="bottom" eqnalign="indent"><?TeX \begin{equation}
\eqalign{
\fl
\chit_{2k-1,b}(q)=
\ch_{2k-1,b}(q)+\bigl(1-\delta_{b,3}\delta_{k,1}\bigr)\ch_{2k+1,b}(q),\tqs
b=1,2,3\\
\fl \chit_{a,3k-b}(q)=\ch_{a,3k-b}(q)
+\bigl(1-\delta_{a,2}\delta_{k,1}\bigr)\ch_{a,3k+b}(q),\tqs a,b=1,2\\
\fl \chit_{3,3k+b}(q)=\ch_{1,3k-3+b}(q)+\ch_{1,3k+b}(q)+\ch_{1,3k+3-b}(q)\\
+\ \ch_{1,3k+3+b}(q)+\ch_{1,3k+6-b}(q)+\ch_{1,3k+9-b}(q),\tqs b=1,2}
\label{chitch}
\end{equation} ?><eqn-graphic format="GIF" filename="images/jstat256772eqn8.gif" position="indent" align="middle"/></display-eqn>
jurer <inline-eqn notation="TeX" type="simple" font="italic"><?TeX k\in \mathbb {N} ?><inline-graphic format="GIF" filename="images/jstat256772ieqn6.gif" align="middle"/></inline-eqn>. Gur qrpbzcbfvgvba va gur trareny pnfr vf qvfphffrq va gur nccraqvk bs [<cite linkend="jstat256772bib43" show="author" override="no">56</cite>].
</p>
      </sec-level2>
    </sec-level1>
    <sec-level1 id="jstat256772s2" label="2" appendix="no" type="num" style="arabic" inherit="yes">
      <heading>Shfvba nytroenf</heading>
      <p indent="no" type="new-line">Gur <italic>shaqnzragny</italic> shfvba nytroen <inline-eqn notation="TeX" type="simple" font="italic"><?TeX \bigl \langle (2,1), (1,2)\bigr \rangle
?><inline-graphic format="GIF" filename="images/jstat256772ieqn10.gif" align="middle"/></inline-eqn> vf qrsvarq nf gur shfvba nytroen trarengrq ol gur shaqnzragny ercerfragngvbaf
(7,8) 
naq (8,7). Jr svaq gung pybfher bs guvf shfvba nytroen erdhverf gur vapyhfvba bs n inevrgl bs bgure
ercerfragngvbaf
<display-eqn id="jstat256772eqn9" textype="eqnarray" eqnnum="2.1" notation="TeX" lines="single" type="complex" font="italic" number="yes" numalign="bottom" eqnalign="indent"><?TeX \begin{eqnarray}
\fl
\bigl\langle(2,1), (1,2)\bigr\rangle=
\bigl\langle(1,1), (1,2), (2k,a), (1,3k), (2k,3),
\R_{2k,a}^{1,0}, \R_{2k,3}^{1,0}, \R_{a,3k}^{0,b},
\R_{2k,3}^{1,b}; \nonumber\\
a,b=1,2; k\in\mathbb{N}\bigr\rangle
\label{A2112}
\end{eqnarray} ?><eqn-graphic format="GIF" filename="images/jstat256772eqn9.gif" position="indent" align="middle"/></display-eqn>
gb or qvfphffrq arkg.
</p>
      <sec-level2 id="jstat256772s2.1" label="2.1" appendix="no" type="num" style="arabic" inherit="yes">
        <heading>Vaqrpbzcbfnoyr ercerfragngvbaf bs enax 7 be 6</heading>
        <p indent="no" type="new-line">Sbe <inline-eqn notation="TeX" type="simple" font="italic"><?TeX k\in \mathbb {N} ?><inline-graphic format="GIF" filename="images/jstat256772ieqn6.gif" align="middle"/></inline-eqn>, gur ercerfragngvbaf qrabgrq ol <inline-eqn notation="TeX" type="simple" font="italic"><?TeX \R_{2k,1}^{1,0} ?><inline-graphic format="GIF" filename="images/jstat256772ieqn11.gif" align="middle"/></inline-eqn>, <inline-eqn notation="TeX" type="simple" font="italic"><?TeX \R_{2k,2}^{1,0} ?><inline-graphic format="GIF" filename="images/jstat256772ieqn12.gif" align="middle"/></inline-eqn>, <inline-eqn notation="TeX" type="simple" font="italic"><?TeX \R_{2k,3}^{1,0} ?><inline-graphic format="GIF" filename="images/jstat256772ieqn13.gif" align="middle"/></inline-eqn>, <inline-eqn notation="TeX" type="simple" font="italic"><?TeX \R_{1,3k}^{0,1} ?><inline-graphic format="GIF" filename="images/jstat256772ieqn14.gif" align="middle"/></inline-eqn>, <inline-eqn notation="TeX" type="simple" font="italic"><?TeX \R_{1,3k}^{0,2} ?><inline-graphic format="GIF" filename="images/jstat256772ieqn15.gif" align="middle"/></inline-eqn>, <inline-eqn notation="TeX" type="simple" font="italic"><?TeX \R_{2,3k}^{0,1} ?><inline-graphic format="GIF" filename="images/jstat256772ieqn16.gif" align="middle"/></inline-eqn> naq <inline-eqn notation="TeX" type="simple" font="italic"><?TeX \R_{2,3k}^{0,2} ?><inline-graphic format="GIF" filename="images/jstat256772ieqn17.gif" align="middle"/></inline-eqn> ner vaqrpbzcbfnoyr ercerfragngvbaf bs enax 7, juvyr <inline-eqn notation="TeX" type="simple" font="italic"><?TeX \R_{2k,3}^{1,1} ?><inline-graphic format="GIF" filename="images/jstat256772ieqn18.gif" align="middle"/></inline-eqn> naq <inline-eqn notation="TeX" type="simple" font="italic"><?TeX \R_{2k,3}^{1,2} ?><inline-graphic format="GIF" filename="images/jstat256772ieqn19.gif" align="middle"/></inline-eqn> ner vaqrpbzcbfnoyr ercerfragngvbaf bs enax 6. Gurve punenpgref ernq
<display-eqn id="jstat256772eqn10" textype="equation" eqnnum="2.2" notation="TeX" lines="single" type="complex" font="italic" number="yes" numalign="bottom" eqnalign="indent"><?TeX \begin{equation}
\eqalign{
\fl
\chit[\R_{2k,b}^{1,0}](q)=\chit_{2k-1,b}(q)+\chit_{2k+1,b}(q)\\
\eql \bigl(1-\delta_{b,3}\delta_{k,1}\bigr)\ch_{2k-1,b}(q)+2\ch_{2k+1,b}(q)
+\ch_{2k+3,b}(q),\tqs b=1,2,3\\
\fl
\chit[\R_{a,3k}^{0,b}](q)=\chit_{a,3k-b}(q)+\chit_{a,3k+b}(q)\\
\eql \bigl(1-\delta_{a,2}\delta_{k,1}\bigr)\ch_{a,3k-b}(q)+2\ch_{a,3k+b}(q)
+\ch_{a,3(k+2)-b}(q),\tqs a,b=1,2\\
\fl
\chit[\R_{2k,3}^{1,b}](q)=\chit_{2k-1,3-b}(q)+\chit_{2k-1,3+b}(q)
+\chit_{2k+1,3-b}(q)+\chit_{2k+1,3+b}(q)\\
\eql \bigl(1-\delta_{k,1}\bigr)\ch_{1,3k-3-b}(q)+2
\bigl(1-\delta_{k,1}\bigr)\ch_{1,3k-3+b}(q)
+2\ch_{1,3k-b}(q)\\
+\ 4\ch_{1,3k+b}(q)+\bigl(2-\delta_{k,1}\bigr)\ch_{1,3k+3-b}(q)
+2\ch_{1,3k+3+b}(q)\\
+\ 2\ch_{1,3k+6-b}(q)+\ch_{1,3k+9-b}(q),\tqs b=1,2}
\label{chiR}
\end{equation} ?><eqn-graphic format="GIF" filename="images/jstat256772eqn10.gif" position="indent" align="middle"/></display-eqn>
vaqvpngvat gung bar znl pbafvqre gur vaqrpbzcbfnoyr ercerfragngvbaf nf ‘vaqrpbzcbfnoyr
pbzovangvbaf’ bs Xnp ercerfragngvbaf. Gur cnegvpvcngvat Xnp ercerfragngvbaf ner bs pbhefr
gur barf jubfr punenpgref nccrne va (<eqnref linkend="jstat256772eqn10" override="no">7.7</eqnref>). Va gur pnfr bs gur vaqrpbzcbfnoyr ercerfragngvba
<inline-eqn notation="TeX" type="simple" font="italic"><?TeX \R_{2k,b}^{1,0} ?><inline-graphic format="GIF" filename="images/jstat256772ieqn20.gif" align="middle"/></inline-eqn> (be <inline-eqn notation="TeX" type="simple" font="italic"><?TeX \R_{a,3k}^{0,b} ?><inline-graphic format="GIF" filename="images/jstat256772ieqn21.gif" align="middle"/></inline-eqn>) bs enax 7, bhe ynggvpr nanylfvf vaqvpngrf gung n Wbeqna pryy vf sbezrq orgjrra rirel fgngr va
pu<sub valign="no">7<italic>x</italic>+8,<italic>o</italic></sub>(<italic>d</italic>) (be
pu<sub valign="no"><italic>n</italic>,6<italic>x</italic>+<italic>o</italic></sub>(<italic>d</italic>)) naq vgf cnegare fgngr
va gur frpbaq pbcl bs pu<sub valign="no">7<italic>x</italic>+8,<italic>o</italic></sub>(<italic>d</italic>) 
(be pu<sub valign="no"><italic>n</italic>,6<italic>x</italic>+<italic>o</italic></sub>(<italic>d</italic>)), naq abjurer ryfr. Va gur pnfr bs gur vaqrpbzcbfnoyr ercerfragngvba <inline-eqn notation="TeX" type="simple" font="italic"><?TeX \R_{2k,3}^{1,b} ?><inline-graphic format="GIF" filename="images/jstat256772ieqn22.gif" align="middle"/></inline-eqn> bs enax 6, bhe ynggvpr nanylfvf vaqvpngrf gung sbe rirel dhnegrg bs zngpuvat fgngrf va gur sbhe pbcvrf
bs pu<sub valign="no">8,6<italic>x</italic>+<italic>o</italic></sub>(<italic>d</italic>), n enax-6 Wbeqna pryy vf sbezrq nybat jvgu n fvatyr fgngr. Vg yvxrjvfr nccrnef gung n Wbeqna
pryy bs enax 7 vf sbezrq orgjrra rirel cnve bs zngpuvat fgngrf va gur veerqhpvoyr pbzcbaragf
jvgu zhygvcyvpvgl 7.</p>
        <p indent="yes" type="new-line">Gur abgngvba <inline-eqn notation="TeX" type="simple" font="italic"><?TeX \R_{r,s}^{a,b} ?><inline-graphic format="GIF" filename="images/jstat256772ieqn23.gif" align="middle"/></inline-eqn> vf zrnag gb ersyrpg fvzcyr cebcregvrf bs gur uvture-enax vaqrpbzcbfnoyr ercerfragngvbaf.
Gur cnve bs ybjre vaqvprf guhf ersref gb n ‘flzzrgel cbvag’ va gur Xnp gnoyr nebhaq juvpu
na vaqrpbzcbfnoyr pbzovangvba bs Xnp ercerfragngvbaf ner ybpngrq. Gur cnve bs hccre
vaqvprf vaqvpngrf gur qvfgevohgvba bs gurfr ercerfragngvbaf bs juvpu gurer ner rvgure gjb (vs
<italic>n</italic> = 9 be
<italic>o</italic> = 9) be sbhe (vs <inline-eqn notation="TeX" type="simple" font="italic"><?TeX a,b\neq 0 ?><inline-graphic format="GIF" filename="images/jstat256772ieqn24.gif" align="middle"/></inline-eqn>). Gurve ybpngvbaf pbeerfcbaq gb raqcbvagf be pbearef, erfcrpgviryl, bs n yvar frtzrag be n erpgnatyr jvgu
prager ng (<italic>e</italic>,<italic>f</italic>). Guvf fgehpgher vf rapbqrq arngyl va gur punenpgre rkcerffvbaf (<eqnref linkend="jstat256772eqn10" override="no">7.7</eqnref>).</p>
        <p indent="yes" type="new-line">Vg sbyybjf sebz gur ynggvpr gung gur shaqnzragny shfvba nytroen znl or qrfpevorq
ol frcnengvat gur ercerfragngvbaf vagb n ubevmbagny naq n iregvpny cneg. Orsber
qvfphffvat vzcyvpngvbaf bs guvf, jr rknzvar gur gjb qverpgvbaf vaqvivqhnyyl, naq
vagebqhpr fbzr nooerivngvbaf. Gb pbzcnpgvsl gur shfvba ehyrf, jr hfr gur abgngvba
<display-eqn id="jstat256772eqn11" textype="equation" eqnnum="2.3" notation="TeX" lines="single" type="complex" font="italic" number="yes" numalign="bottom" eqnalign="indent"><?TeX \begin{equation}
(r,-s) \equiv (-r,s) \equiv-(r,s),\tqs
\R_{-r,s}^{a,b} \equiv \R_{r,-s}^{a,b}
\equiv-\R_{r,s}^{a,b}
\end{equation} ?><eqn-graphic format="GIF" filename="images/jstat256772eqn11.gif" position="indent" align="middle"/></display-eqn>
vzcylvat, va cnegvphyne, gung <inline-eqn notation="TeX" type="simple" font="italic"><?TeX (0,s)\equiv (r,0)\equiv \R_{0,s}^{a,b}\equiv \R
_{r,0}^{a,b}\equiv 0 ?><inline-graphic format="GIF" filename="images/jstat256772ieqn25.gif" align="middle"/></inline-eqn>, naq qrsvar gur Xebarpxre qrygn pbzovangvbaf
<display-eqn id="jstat256772eqn12" textype="equation" eqnnum="2.4" notation="TeX" lines="single" type="complex" font="italic" number="yes" numalign="bottom" eqnalign="indent"><?TeX \begin{equation}
\eqalign{
\fl \dkk=2-\delta_{j,|k-k'|}-\delta_{j,k+k'} \\
\fl
\ddkk=4-3\delta_{j,|k-k'|-1}-2\delta_{j,|k-k'|}-\delta_{j,|k-k'|+1}-\delta_{j,k+k'-1}-2\delta_{j,k+k'}-3\delta_{j,k+k'+1}\\
\fl
\dddkk=8-7\delta_{j,|k-k'|-2}-6\delta_{j,|k-k'|-1}-4\delta_{j,|k-k'|}-2\delta_{j,|k-k'|+1}-\delta_{j,|k-k'|+2}\\-\ \delta_{j,k+k'-2}-2\delta_{j,k+k'-1}-4\delta_{j,k+k'}-6\delta_{j,k+k'+1}-7\delta_{j,k+k'+2}.}
\label{d24}
\end{equation} ?><eqn-graphic format="GIF" filename="images/jstat256772eqn12.gif" position="indent" align="middle"/></display-eqn>
</p>
      </sec-level2>
      <sec-level2 id="jstat256772s2.2" label="2.2" appendix="no" type="num" style="arabic" inherit="yes">
        <heading>Ubevmbagny shfvba nytroen</heading>
        <p indent="no" type="new-line">Gur <italic>ubevmbagny</italic> shfvba nytroen <inline-eqn notation="TeX" type="simple" font="italic"><?TeX \bigl \langle (2,1)\bigr \rangle ?><inline-graphic format="GIF" filename="images/jstat256772ieqn26.gif" align="middle"/></inline-eqn> vf qrsvarq nf gur shfvba nytroen trarengrq ol gur shaqnzragny ercerfragngvba
(7,8). Jr svaq gung pybfher bs guvf shfvba nytroen erdhverf gur vapyhfvba bs gur Xnp ercerfragngvbaf
(7<italic>x</italic>,8) 
naq gur enax-7 vaqrpbzcbfnoyr ercerfragngvbaf <inline-eqn notation="TeX" type="simple" font="italic"><?TeX \R_{2k,1}^{1,0} ?><inline-graphic format="GIF" filename="images/jstat256772ieqn11.gif" align="middle"/></inline-eqn>
<display-eqn id="jstat256772eqn13" textype="equation" eqnnum="2.5" notation="TeX" lines="single" type="complex" font="italic" number="yes" numalign="bottom" eqnalign="indent"><?TeX \begin{equation}
\langle(2,1)\rangle=\langle(2k,1), \R_{2k,1}^{1,0}; k\in\mathbb{N}
\rangle.\label{A21}
\end{equation} ?><eqn-graphic format="GIF" filename="images/jstat256772eqn13.gif" position="indent" align="middle"/></display-eqn>
Jr pbawrpgher gung gur shfvba nytroen <inline-eqn notation="TeX" type="simple" font="italic"><?TeX \bigl \langle (2,1)\bigr \rangle
?><inline-graphic format="GIF" filename="images/jstat256772ieqn26.gif" align="middle"/></inline-eqn> ernqf
<display-eqn id="jstat256772eqn14" textype="equation" eqnnum="2.6" notation="TeX" lines="single" type="complex" font="italic" number="yes" numalign="bottom" eqnalign="indent"><?TeX \begin{equation}
\eqalign{
(2k,1)\otimes(2k',1)=\bigoplus_{j=|k-k'|+1, \mathrm{by} 2}^{k+k'-1}
\R_{2j,1}^{1,0}\\
(2k,1)\otimes \R_{2k',1}^{1,0}=\bigoplus_{j=|k-k'|}^{k+k'}
\dkk(2j,1)\\
\R_{2k,1}^{1,0}\otimes \R_{2k',1}^{1,0}=\bigoplus_{j=|k-k'|}^{k+k'}
\dkk\R_{2j,1}^{1,0}.}
\label{fusion21}
\end{equation} ?><eqn-graphic format="GIF" filename="images/jstat256772eqn14.gif" position="indent" align="middle"/></display-eqn>
Guvf shfvba nytroen qbrf abg pbagnva na vqragvgl.
</p>
      </sec-level2>
      <sec-level2 id="jstat256772s2.3" label="2.3" appendix="no" type="num" style="arabic" inherit="yes">
        <heading>Iregvpny shfvba nytroen</heading>
        <p indent="no" type="new-line">Gur <italic>iregvpny</italic> shfvba nytroen <inline-eqn notation="TeX" type="simple" font="italic"><?TeX \bigl \langle (1,2)\bigr \rangle ?><inline-graphic format="GIF" filename="images/jstat256772ieqn27.gif" align="middle"/></inline-eqn> vf qrsvarq nf gur shfvba nytroen trarengrq ol gur shaqnzragny ercerfragngvba
(8,7). Jr svaq gung pybfher bs guvf shfvba nytroen erdhverf gur vapyhfvba bs gur Xnp ercerfragngvbaf
(8,8) 
naq (8,6<italic>x</italic>) 
naq gur enax-7 vaqrpbzcbfnoyr ercerfragngvbaf <inline-eqn notation="TeX" type="simple" font="italic"><?TeX \R_{2k,1}^{0,b} ?><inline-graphic format="GIF" filename="images/jstat256772ieqn28.gif" align="middle"/></inline-eqn>
<display-eqn id="jstat256772eqn15" textype="equation" eqnnum="2.7" notation="TeX" lines="single" type="complex" font="italic" number="yes" numalign="bottom" eqnalign="indent"><?TeX \begin{equation}
\bigl\langle(1,2)\bigr\rangle
=\bigl\langle(1,1), (1,2), (1,3k), \R_{1,3k}^{0,b}; b=1,2;
k\in\mathbb{N}\bigr\rangle.\label{A12}
\end{equation} ?><eqn-graphic format="GIF" filename="images/jstat256772eqn15.gif" position="indent" align="middle"/></display-eqn>
Yrggvat <italic>K</italic> 
qrabgr nal bs gurfr ercerfragngvbaf, jr pbawrpgher gung gur shfvba nytroen <inline-eqn notation="TeX" type="simple" font="italic"><?TeX \bigl \langle
(1,2)\bigr \rangle ?><inline-graphic format="GIF" filename="images/jstat256772ieqn27.gif" align="middle"/></inline-eqn> ernqf
<display-eqn id="jstat256772eqn16" textype="eqnarray" eqnnum="2.8" notation="TeX" lines="single" type="complex" font="italic" number="yes" numalign="bottom" eqnalign="indent"><?TeX \begin{eqnarray}
\fl
(1,1)\otimes X=X\nonumber\\
\fl
(1,2)\otimes (1,2)=(1,1)\oplus(1,3)\nonumber\\
\fl
(1,2)\otimes (1,3k)=\R_{1,3k}^{0,1}\nonumber\\
\fl
(1,2)\otimes \R_{1,3k}^{0,1}=\R_{1,3k}^{0,2}\oplus2(1,3k)\nonumber\\
\fl
(1,2)\otimes \R_{1,3k}^{0,2}=\R_{1,3k}^{0,1}\oplus(1,3(k-1))\oplus
(1,3(k+1))\nonumber\\
\fl
(1,3k)\otimes(1,3k')=
\bigoplus_{j=|k-k'|+1, \mathrm{by} 2}^{k+k'-1}\bigl(\R_{1,3j}^{0,2}\oplus(1,3j)\bigr)\nonumber\\
\fl
(1,3k)\otimes \R_{1,3k'}^{0,1}=
\Bigl(\bigoplus_{j=|k-k'|+1, \mathrm{by} 2}^{k+k'-1}2\R_{1,3j}^{0,1}\Bigr)
\oplus\Bigl(\bigoplus_{j=|k-k'|, \mathrm{by} 2}^{k+k'}\dkk
(1,3j)\Bigr)\nonumber\\
\fl
(1,3k)\otimes \R_{1,3k'}^{0,2}=
\Bigl(\bigoplus_{j=|k-k'|, \mathrm{by} 2}^{k+k'}\dkk\R_{1,3j}^{0,1}\Bigr)
\oplus\Bigl(\bigoplus_{j=|k-k'|+1, \mathrm{by} 2}^{k+k'-1}2(1,3j)\Bigr)\nonumber\\
\fl
\R_{1,3k}^{0,1}\otimes \R_{1,3k'}^{0,1}=
\Bigl(\bigoplus_{j=|k-k'|, \mathrm{by} 2}^{k+k'}\dkk\R_{1,3j}^{0,1}\Bigr)
\oplus\Bigl(\bigoplus_{j=|k-k'|+1, \mathrm{by} 
2}^{k+k'-1}\bigl(2\R_{1,3j}^{0,2}\oplus4(1,3j)\bigr)\Bigr)\nonumber\\
\fl
\R_{1,3k}^{0,1}\otimes \R_{1,3k'}^{0,2}=
\Bigl(\bigoplus_{j=|k-k'|+1, \mathrm{by} 2}^{k+k'-1}2\R_{1,3j}^{0,1}\Bigr)
\oplus
\Bigl(\bigoplus_{j=|k-k'|, \mathrm{by} 2}^{k+k'}\dkk\bigl(\R_{1,3j}^{0,2}
\oplus2(1,3j)\bigr)\Bigr)\nonumber\\
\fl
\R_{1,3k}^{0,2}\otimes R_{1,3k'}^{0,2}=
\Bigl(\bigoplus_{j=|k-k'|, \mathrm{by} 2}^{k+k'}\dkk\R_{1,3j}^{0,1}\Bigr)
\oplus\Bigl(\bigoplus_{j=|k-k'|+1, \mathrm{by} 2}^{k+k'-1}2\R_{1,3j}^{0,2}\Bigr)\nonumber\\
\oplus\ \Bigl(\bigoplus_{j=|k-k'|-1, \mathrm{by} 2}^{k+k'+1}
\ddkk(1,3j)\Bigr).\label{fusion12}
\end{eqnarray} ?><eqn-graphic format="GIF" filename="images/jstat256772eqn16.gif" position="indent" align="middle"/></display-eqn>
Vg vf abgrq gung sbe <italic>w</italic> = |<italic>x</italic>−<italic>x</italic>′|−8 
(zbq 7), nf va <inline-eqn notation="TeX" type="simple" font="italic"><?TeX \R_{1,3k}^{0,2}\otimes R_{1,3k'}^{0,2} ?><inline-graphic format="GIF" filename="images/jstat256772ieqn29.gif" align="middle"/></inline-eqn>, gur shfvba zhygvcyvpvgl δ<sub valign="no"><italic>w</italic>,{<italic>x</italic>,<italic>x</italic>′}</sub><sup>(5)</sup> 
erqhprf gb 5−6δ<sub valign="no"><italic>w</italic>,|<italic>x</italic>−<italic>x</italic>′|−8</sub>−δ<sub valign="no"><italic>w</italic>,|<italic>x</italic>−<italic>x</italic>′|+8</sub>−δ<sub valign="no"><italic>w</italic>,<italic>x</italic>+<italic>x</italic>′−8</sub>−6δ<sub valign="no"><italic>w</italic>,<italic>x</italic>+<italic>x</italic>′+8</sub>. Gur
ercerfragngvba (8,8) 
vf gur vqragvgl bs guvf iregvpny shfvba nytroen.
</p>
      </sec-level2>
      <sec-level2 id="jstat256772s2.4" label="2.4" appendix="no" type="num" style="arabic" inherit="yes">
        <heading>Pbzcnevfba jvgu Ernq naq Fnyrhe</heading>
        <p indent="no" type="new-line">Vg vf irevsvrq gung
<display-eqn id="jstat256772eqn17" textype="equation" eqnnum="2.9" notation="TeX" lines="single" type="complex" font="italic" number="yes" numalign="bottom" eqnalign="indent"><?TeX \begin{equation}
\bigl\langle(1,1),(1,6k-3),\R_{1,6k}^{0,1},
\R_{1,6k-3}^{0,2}; k\in\mathbb{N}\bigr\rangle
\label{subvert}
\end{equation} ?><eqn-graphic format="GIF" filename="images/jstat256772eqn17.gif" position="indent" align="middle"/></display-eqn>
vf n fhonytroen bs gur iregvpny shfvba nytroen. Vg pbeerfcbaqf gb gur shfvba nytroen bs pevgvpny
crepbyngvba qvfphffrq ol Ernq naq Fnyrhe va [<cite linkend="jstat256772bib48" show="author" override="no">53</cite>]. Gb nccerpvngr guvf, jr
cebivqr n qvpgvbanel sbe genafyngvat gur ercerfragngvbaf trarengvat gur fhonytroen (<eqnref linkend="jstat256772eqn17" override="no">7.4</eqnref>) vagb
gur abgngvba hfrq va [<cite linkend="jstat256772bib48" show="author" override="no">53</cite>]
<display-eqn id="jstat256772eqn18" textype="equation" eqnnum="2.10" notation="TeX" lines="single" type="complex" font="italic" number="yes" numalign="bottom" eqnalign="indent"><?TeX \begin{equation}
\eqalign{
(1,1) \longleftrightarrow \R_0\\
(1,2j+1) \longleftrightarrow \R_j,\tqs j\equiv1 (\mathrm{mod} 3)\\
\R_{1,2j-1}^{0,2} \longleftrightarrow \R_j,\tqs j\equiv2 (\mathrm{mod} 3)\\
\R_{1,2j}^{0,1} \longleftrightarrow \R_j,\tqs j\equiv0 (\mathrm{mod} 3)}
\label{dictRS}
\end{equation} ?><eqn-graphic format="GIF" filename="images/jstat256772eqn18.gif" position="indent" align="middle"/></display-eqn>
jurer <inline-eqn notation="TeX" type="simple" font="italic"><?TeX j\in \mathbb {N} ?><inline-graphic format="GIF" filename="images/jstat256772ieqn30.gif" align="middle"/></inline-eqn>. Jr svaq gung gurve shfvba nytroen vf va nterrzrag jvgu gur fhonytroen (<eqnref linkend="jstat256772eqn17" override="no">7.4</eqnref>) bs gur iregvpny
shfvba nytroen <inline-eqn notation="TeX" type="simple" font="italic"><?TeX \bigl \langle (1,2)\bigr \rangle ?><inline-graphic format="GIF" filename="images/jstat256772ieqn27.gif" align="middle"/></inline-eqn> juvpu vgfrys vf n fhonytroen bs gur shaqnzragny shfvba nytroen <inline-eqn notation="TeX" type="simple" font="italic"><?TeX \bigl \langle
(2,1),(1,2)\bigr \rangle ?><inline-graphic format="GIF" filename="images/jstat256772ieqn10.gif" align="middle"/></inline-eqn> bs pevgvpny crepbyngvba.
</p>
      </sec-level2>
      <sec-level2 id="jstat256772s2.5" label="2.5" appendix="no" type="num" style="arabic" inherit="yes">
        <heading>Shaqnzragny shfvba nytroen</heading>
        <p indent="no" type="new-line">Vg sbyybjf sebz gur ynggvpr qrfpevcgvba gung gur shaqnzragny shfvba nytroen <inline-eqn notation="TeX" type="simple" font="italic"><?TeX \bigl \langle
(2,1),(1,2)\bigr \rangle ?><inline-graphic format="GIF" filename="images/jstat256772ieqn10.gif" align="middle"/></inline-eqn> vf obgu nffbpvngvir naq pbzzhgngvir. Nf nyernql naabhaprq, vg nyfb sbyybjf sebz gur
ynggvpr gung gur ercerfragngvbaf znl or frcnengrq vagb n ubevmbagny naq n iregvpny cneg. Sbe
gur Xnp ercerfragngvbaf, guvf vzcyvrf
<display-eqn id="jstat256772eqn19" textype="equation" eqnnum="2.11" notation="TeX" lines="single" type="complex" font="italic" number="yes" numalign="bottom" eqnalign="indent"><?TeX \begin{equation}
(r,s)=(r,1)\otimes(1,s).
\label{r11s}
\end{equation} ?><eqn-graphic format="GIF" filename="images/jstat256772eqn19.gif" position="indent" align="middle"/></display-eqn>
Sbe gur checbfrf bs rknzvavat gur shaqnzragny shfvba nytroen, jr vagebqhpr gur
ercerfragngvbaf
<display-eqn id="jstat256772eqn20" textype="equation" eqnnum="2.12" notation="TeX" lines="single" type="complex" font="italic" number="yes" numalign="bottom" eqnalign="indent"><?TeX \begin{equation}
\eqalign{ (2k,3k')=(2k,1)\otimes(1,3k'),\tqs
\R_{2k,3k'}^{1,0}=\R_{2k,1}^{1,0}\otimes(1,3k')\\
\R_{2k,3k'}^{0,b}=(2k,1)\otimes \R_{1,3k'}^{0,b},\tqs
\R_{2k,3k'}^{1,b}=\R_{2k,1}^{1,0}\otimes \R_{1,3k'}^{0,b}}
\label{keqnk}
\end{equation} ?><eqn-graphic format="GIF" filename="images/jstat256772eqn20.gif" position="indent" align="middle"/></display-eqn>
guhf qrsvarq nf gur erfhyg bs pregnva fvzcyr shfvbaf bs ‘n ubevmbagny naq n iregvpny
ercerfragngvba’. Nf jr jvyy fubj ryfrjurer, gurfr ercerfragngvbaf znl or qrpbzcbfrq va
grezf bs gur ercerfragngvbaf yvfgrq va (<eqnref linkend="jstat256772eqn9" override="no">7.8</eqnref>)
<display-eqn id="jstat256772eqn21" textype="equation" eqnnum="2.13" notation="TeX" lines="single" type="complex" font="italic" number="yes" numalign="bottom" eqnalign="indent"><?TeX \begin{equation}
\eqalign{
(2k,3k')=\bigoplus_{j=|k-k'|+1, \mathrm{by} 2}^{k+k'-1}(2j,3),\tqs
\R_{2k,3k'}^{1,0}=\bigoplus_{j=|k-k'|+1, \mathrm{by} 2}^{k+k'-1}\R_{2j,3}^{1,0}\\
\R_{2k,3k'}^{0,b}=\bigoplus_{j=|k-k'|+1, \mathrm{by} 
2}^{k+k'-1}\R_{2,3j}^{0,b},\tqs
\R_{2k,3k'}^{1,b}=\bigoplus_{j=|k-k'|+1, \mathrm{by} 2}^{k+k'-1}\R_{2j,3}^{1,b}
}
\label{keqnkexp}
\end{equation} ?><eqn-graphic format="GIF" filename="images/jstat256772eqn21.gif" position="indent" align="middle"/></display-eqn>
jvgu
<display-eqn id="jstat256772eqn22" textype="equation" eqnnum="2.14" notation="TeX" lines="single" type="complex" font="italic" number="yes" numalign="bottom" eqnalign="indent"><?TeX \begin{equation}
\fl
(2k,3k')=(2k',3k),\quad \R_{2k,3k'}^{1,0}=\R_{2k',3k}^{1,0},\quad
\R_{2k,3k'}^{0,b}=\R_{2k',3k}^{0,b},\quad
\R_{2k,3k'}^{1,b}=\R_{2k',3k}^{1,b}
\label{RequalR}
\end{equation} ?><eqn-graphic format="GIF" filename="images/jstat256772eqn22.gif" position="indent" align="middle"/></display-eqn>
nf fcrpvny vqragvsvpngvbaf rkgraqvat gur frg (<eqnref linkend="jstat256772eqn7" override="no">8.2</eqnref>). Gur shaqnzragny shfvba nytroen vf abj
bognvarq ol fvzcyl nccylvat (<eqnref linkend="jstat256772eqn20" override="no">7.87</eqnref>) naq (<eqnref linkend="jstat256772eqn21" override="no">7.86</eqnref>) gb gur shfvba bs n cnve bs ercerfragngvbaf
va (<eqnref linkend="jstat256772eqn9" override="no">7.8</eqnref>). Jr vyyhfgengr guvf jvgu n trareny ohg fbzrjung sbezny rinyhngvba jurer jr yrg
<inline-eqn notation="TeX" type="simple" font="italic"><?TeX A_{r,s}=\bar {a}_{r,1}\otimes a_{1,s} ?><inline-graphic format="GIF" filename="images/jstat256772ieqn31.gif" align="middle"/></inline-eqn>, <inline-eqn notation="TeX" type="simple" font="italic"><?TeX B_{r',s'}=\bar {b}_{r',1}\otimes b_{1,s'} ?><inline-graphic format="GIF" filename="images/jstat256772ieqn32.gif" align="middle"/></inline-eqn>, <inline-eqn notation="TeX" type="simple" font="italic"><?TeX \bar {a}_{r,1}\otimes \bar {b}_{r',1}=\bigoplus_{r''}\bar {c}_{r'',1} ?><inline-graphic format="GIF" filename="images/jstat256772ieqn33.gif" align="middle"/></inline-eqn> naq <inline-eqn notation="TeX" type="simple" font="italic"><?TeX a_{1,s}\otimes b_{1,s'}=\bigoplus_{s''}c_{1,s''} ?><inline-graphic format="GIF" filename="images/jstat256772ieqn34.gif" align="middle"/></inline-eqn>. Bhe shfvba cerfpevcgvba abj lvryqf
<display-eqn id="jstat256772eqn23" textype="eqnarray" eqnnum="2.15" notation="TeX" lines="single" type="complex" font="italic" number="yes" numalign="bottom" eqnalign="indent"><?TeX \begin{eqnarray}
\fl
A_{r,s}\otimes B_{r',s'}=\Bigl(\bar{a}_{r,1}\otimes a_{1,s}\Bigr)\otimes
\Bigl(\bar{b}_{r',1}\otimes b_{1,s'}\Bigr)
=\Bigl(\bar{a}_{r,1}\otimes\bar{b}_{r',1}\Bigr)\otimes
\Bigl(a_{1,s}\otimes b_{1,s'}\Bigr)\nonumber\\
\eql \Bigl(\bigoplus_{r''}\bar{c}_{r'',1}\Bigr)\otimes\Bigl(\bigoplus_{s''}c_{1,s''}\Bigr)
=\bigoplus_{r'',s''}C_{r'',s''}
\label{rs}
\end{eqnarray} ?><eqn-graphic format="GIF" filename="images/jstat256772eqn23.gif" position="indent" align="middle"/></display-eqn>
jurer <inline-eqn notation="TeX" type="simple" font="italic"><?TeX C_{r'',s''}=\bar {c}_{r'',1}\otimes c_{1,s''} ?><inline-graphic format="GIF" filename="images/jstat256772ieqn35.gif" align="middle"/></inline-eqn>. Hfvat guvf, gur shaqnzragny shfvba nytroen <inline-eqn notation="TeX" type="simple" font="italic"><?TeX \bigl \langle (2,1),(1,2)\bigr \rangle
?><inline-graphic format="GIF" filename="images/jstat256772ieqn10.gif" align="middle"/></inline-eqn> sbyybjf fgenvtugsbejneqyl sebz gur shfvba nytroenf <inline-eqn notation="TeX" type="simple" font="italic"><?TeX \bigl \langle (2,1)\bigr \rangle
?><inline-graphic format="GIF" filename="images/jstat256772ieqn26.gif" align="middle"/></inline-eqn> naq <inline-eqn notation="TeX" type="simple" font="italic"><?TeX \bigl \langle (1,2)\bigr \rangle ?><inline-graphic format="GIF" filename="images/jstat256772ieqn27.gif" align="middle"/></inline-eqn> gbtrgure jvgu (<eqnref linkend="jstat256772eqn20" override="no">7.87</eqnref>) naq (<eqnref linkend="jstat256772eqn21" override="no">7.86</eqnref>). Va cnegvphyne, vg sbyybjf ernqvyl gung gur Xnp ercerfragngvba
(8,8) 
vf gur <italic>vqragvgl</italic> bs gur shaqnzragny shfvba nytroen <inline-eqn notation="TeX" type="simple" font="italic"><?TeX \bigl \langle (2,1),(1,2)\bigr
\rangle ?><inline-graphic format="GIF" filename="images/jstat256772ieqn10.gif" align="middle"/></inline-eqn>.</p>
        <p indent="yes" type="new-line">Va guvf oevrs pbzzhavpngvba, jr jvyy bayl nccyl guvf shfvba cerfpevcgvba rkcyvpvgyl gb gur
shfvba bs gur gjb enax-7 vaqrpbzcbfnoyr ercerfragngvbaf <inline-eqn notation="TeX" type="simple" font="italic"><?TeX \R_{2k,2}^{1,0} ?><inline-graphic format="GIF" filename="images/jstat256772ieqn12.gif" align="middle"/></inline-eqn> naq <inline-eqn notation="TeX" type="simple" font="italic"><?TeX \R_{2,3k'}^{0,2} ?><inline-graphic format="GIF" filename="images/jstat256772ieqn36.gif" align="middle"/></inline-eqn>
<display-eqn id="jstat256772eqn24" textype="eqnarray" eqnnum="2.16" notation="TeX" lines="single" type="complex" font="italic" number="yes" numalign="bottom" eqnalign="indent"><?TeX \begin{eqnarray}
\fl \R_{2k,2}^{1,0}\otimes\R_{2,3k'}^{0,2}=
\Bigl(\R_{2k,1}^{1,0}\otimes (1,2)\Bigr)
\otimes\Bigl((2,1)\otimes \R_{1,3k'}^{0,2}\Bigr)\nonumber\\
\eql \Bigl(\R_{2k,1}^{1,0}\otimes (2,1)\Bigr)\otimes\Bigl((1,2)\otimes
\R_{1,3k'}^{0,2}\Bigr)\nonumber\\
\eql \Bigl((2(k-1),1)\oplus2(2k,1)\oplus(2(k+1),1)\Bigr)\nonumber\\
\otimes\ \Bigl(\R_{1,3k'}^{0,1}
\oplus
(1,3(k'-1))\oplus(1,3(k'+1))\Bigr)\nonumber\\
\eql \Bigl(\bigoplus_{j=|k-k'|}^{k+k'}\dkk\R_{2,3j}^{0,1}\Bigr)
\oplus\Bigl(\bigoplus_{j=|k-k'|-1}^{k+k'+1}\delta_{j,\{k,k'\}}^{(4)}(2j,3)\Bigr)
\label{ex22}
\end{eqnarray} ?><eqn-graphic format="GIF" filename="images/jstat256772eqn24.gif" position="indent" align="middle"/></display-eqn>
naq gb gur shfvba bs gjb enax-6 vaqrpbzcbfnoyr ercerfragngvbaf
<display-eqn id="jstat256772eqn25" textype="eqnarray" eqnnum="2.17" notation="TeX" lines="single" type="complex" font="italic" number="yes" numalign="bottom" eqnalign="indent"><?TeX \begin{eqnarray}
\fl
\R_{2k,3}^{1,1}\otimes \R_{2k',3}^{1,1}=\Bigl(\R_{2k,1}^{1,0}\otimes
\R_{1,3}^{0,1}\Bigr)
\otimes\Bigl(\R_{2k',1}^{1,0}\otimes \R_{1,3}^{0,1}\Bigr)
=\Bigl(\R_{2k,1}^{1,0}\otimes \R_{2k',1}^{1,0}\Bigr)
\otimes\Bigl(\R_{1,3}^{0,1}\otimes \R_{1,3}^{0,1}\Bigr)\nonumber\\
\eql \Bigl(\bigoplus_{j=|k-k'|}^{k+k'}\dkk\R_{2j,1}^{1,0}\Bigr)
\otimes\Bigl(\R_{1,6}^{0,1}\oplus2\R_{1,3}^{0,2}\oplus4(1,3)\Bigr)\nonumber\\
\eql \Bigl(\bigoplus_{j=|k-k'|-1}^{k+k'+1}\ddkk\R_{2j,3}^{1,1}\Bigr)
\oplus\Bigl(\bigoplus_{j=|k-k'|}^{k+k'}\dkk\bigl(2\R_{2j,3}^{1,2}\oplus4\R_{2j,3}^{1,0}\bigr)\Bigr)
\label{ex3311}
\end{eqnarray} ?><eqn-graphic format="GIF" filename="images/jstat256772eqn25.gif" position="indent" align="middle"/></display-eqn>
naq yvxrjvfr
<display-eqn id="jstat256772eqn26" textype="equation" eqnnum="2.18" notation="TeX" lines="single" type="complex" font="italic" number="yes" numalign="bottom" eqnalign="indent"><?TeX \begin{equation}
\eqalign{
\fl
\R_{2k,3}^{1,1}\otimes \R_{2k',3}^{1,2}=
\Bigl(\bigoplus_{j=|k-k'|}^{k+k'}\dkk2\R_{2j,3}^{1,1}\Bigr)
\oplus\Bigl(\bigoplus_{j=|k-k'|-1}^{k+k'+1}\ddkk\bigl(\R_{2j,3}^{1,2}\oplus2\R_{2j,3}^{1,0}\bigr)\Bigr)
\\
\fl
\R_{2k,3}^{1,2}\otimes \R_{2k',3}^{1,2}=
\Bigl(\bigoplus_{j=|k-k'|-1}^{k+k'+1}\ddkk\R_{2j,3}^{1,1}\Bigr)
\oplus\Bigl(\bigoplus_{j=|k-k'|}^{k+k'}\dkk2\R_{2j,3}^{1,2}\Bigr)\nonumber\\
\oplus\Bigl(\bigoplus_{j=|k-k'|-2}^{k+k'+2}\dddkk\R_{2j,3}^{1,0}\Bigr).}
\label{ex3312}
\end{equation} ?><eqn-graphic format="GIF" filename="images/jstat256772eqn26.gif" position="indent" align="middle"/></display-eqn></p>
        <p indent="yes" type="new-line">Frireny fhonytroenf bs gur shaqnzragny shfvba nytroen ner rnfvyl vqragvsvrq.
Na vagrerfgvat rknzcyr vf gur bar trarengrq ol gur frg bs enax-6
vaqrpbzcbfnoyr ercerfragngvbaf naq gur enax-7 vaqrpbzcbfnoyr ercerfragngvbaf
<italic>E</italic><sub valign="no">7<italic>x</italic>,6</sub><sup>8,9</sup>.
Gjb bgure abgrjbegul fhonytroenf ner gur barf trarengrq ol nyy gur ercerfragngvbaf va (<eqnref linkend="jstat256772eqn9" override="no">7.8</eqnref>) rkprcg
(8,7) be
(8,8) 
naq (8,7).</p>
        <p indent="yes" type="new-line">Jr jvfu gb cbvag bhg gung, ng gur yriry bs Xnp punenpgref, gur ubevmbagny, iregvpny naq
shaqnzragny shfvba nytroenf ner nyy pbzcngvoyr jvgu gur <inline-eqn notation="TeX" type="simple" font="italic"><?TeX s\ell (2) ?><inline-graphic format="GIF" filename="images/jstat256772ieqn1.gif" align="middle"/></inline-eqn> fgehpgher
<display-eqn id="jstat256772eqn27" textype="equation" eqnnum="2.19" notation="TeX" lines="single" type="complex" font="italic" number="yes" numalign="bottom" eqnalign="indent"><?TeX \begin{equation}
\phi_n\otimes\phi_{n'}=
\bigoplus_{m=|n-n'|+1, \mathrm{by} 2}^{n+n'-1}\phi_m.
\label{sl2}
\end{equation} ?><eqn-graphic format="GIF" filename="images/jstat256772eqn27.gif" position="indent" align="middle"/></display-eqn>
Guvf vf fgenvtugsbejneq gb rfgnoyvfu sbe gur ubevmbagny naq iregvpny shfvba nytroenf nf
vyyhfgengrq ol gur shfvba <inline-eqn notation="TeX" type="simple" font="italic"><?TeX \R_{2k,1}^{1,0}\otimes \R_{2k',1}^{1,0} ?><inline-graphic format="GIF" filename="images/jstat256772ieqn37.gif" align="middle"/></inline-eqn> jurer (<eqnref linkend="jstat256772eqn27" override="no">7.84</eqnref>) lvryqf
<display-eqn id="jstat256772eqn28" textype="eqnarray" eqnnum="2.20" notation="TeX" lines="single" type="complex" font="italic" number="yes" numalign="bottom" eqnalign="indent"><?TeX \begin{eqnarray}
\fl
\chit[\R_{2k,1}^{1,0}\otimes\R_{2k',1}^{1,0}](q)=
\bigl(\chit_{2k-1,1}(q)+\chit_{2k+1,1}(q)\bigr)\otimes\bigl(\chit_{2k'-1,1}(q)+\chit_{2k'+1,1}(q)\bigr)\nonumber\\
\eql \sum_{j=|2k-2k'|+1, \mathrm{by} 2}^{2(k+k')-3}\chit_{j,1}(q)
+\sum_{j=|2k-2k'-2|+1, \mathrm{by} 2}^{2(k+k')-1}\chit_{j,1}(q)\nonumber\\
+\ \sum_{j=|2k-2k'+2|+1, \mathrm{by} 2}^{2(k+k')-1}\chit_{j,1}(q)
+\sum_{j=|2k-2k'|+1, \mathrm{by} 2}^{2(k+k')+1}\chit_{j,1}(q)\nonumber\\
\eql \sum_{j=|k-k'|}^{k+k'}\dkk\bigl(\chit_{2j-1,1}(q)+\chit_{2j+1,1}(q)\bigr)
\end{eqnarray} ?><eqn-graphic format="GIF" filename="images/jstat256772eqn28.gif" position="indent" align="middle"/></display-eqn>
juvyr
<display-eqn id="jstat256772eqn29" textype="eqnarray" eqnnum="2.21" notation="TeX" lines="single" type="complex" font="italic" number="yes" numalign="bottom" eqnalign="indent"><?TeX \begin{eqnarray}
\chit[\R_{2k,1}^{1,0}\otimes\R_{2k',1}^{1,0}](q) &=
\sum_{j=|k-k'|}^{k+k'}\dkk\chit[\R_{2j,1}^{1,0}](q)\nonumber\\
&=\sum_{j=|k-k'|}^{k+k'}\dkk
\bigl(\chit_{2j-1,1}(q)+\chit_{2j+1,1}(q)\bigr).
\end{eqnarray} ?><eqn-graphic format="GIF" filename="images/jstat256772eqn29.gif" position="indent" align="middle"/></display-eqn>
Gur frcnengvba vagb n ubevmbagny naq n iregvpny cneg (<eqnref linkend="jstat256772eqn19" override="no">7.88</eqnref>) naq (<eqnref linkend="jstat256772eqn20" override="no">7.87</eqnref>) gura vzcyvrf gung
gur punenpgref bs gur shaqnzragny shfvba nytroen rkuvovg gjb vaqrcraqrag <inline-eqn notation="TeX" type="simple" font="italic"><?TeX s\ell
(2) ?><inline-graphic format="GIF" filename="images/jstat256772ieqn1.gif" align="middle"/></inline-eqn> fgehpgherf nf va (<eqnref linkend="jstat256772eqn27" override="no">7.84</eqnref>)—bar va rnpu qverpgvba. Guvf vf pyrneyl erzvavfprag bs gur shfvba
nytroenf bs engvbany (zvavzny) zbqryf jurer gur <inline-eqn notation="TeX" type="simple" font="italic"><?TeX s\ell (2) ?><inline-graphic format="GIF" filename="images/jstat256772ieqn1.gif" align="middle"/></inline-eqn> fgehpgherf ner pneevrq ol gur (punenpgref bs gur) <italic>veerqhpvoyr</italic> ercerfragngvbaf. Urer, ba gur
bgure unaq, gur <inline-eqn notation="TeX" type="simple" font="italic"><?TeX s\ell (2) ?><inline-graphic format="GIF" filename="images/jstat256772ieqn1.gif" align="middle"/></inline-eqn> fgehpgherf ner gvrq gb gur <italic>Xnp</italic> ercerfragngvbaf ohg, qhr gb gur uvture-enax vaqrpbzcbfnoyr
angher bs fbzr bgure ercerfragngvbaf, bayl ng gur yriry bs gurve <italic>punenpgref</italic>.
</p>
      </sec-level2>
      <sec-level2 id="jstat256772s2.6" label="2.6" appendix="no" type="num" style="arabic" inherit="yes">
        <heading>Pbzcnevfba jvgu Roreyr naq Sybue</heading>
        <p indent="no" type="new-line">Gb snpvyvgngr n pbzcnevfba jvgu [<cite linkend="jstat256772bib45" show="author" override="no">50</cite>] ol Roreyr naq Sybue, jr cebivqr n cnegvny
qvpgvbanel eryngvat bhe abgngvba gb gur bar hfrq va [<cite linkend="jstat256772bib45" show="author" override="no">50</cite>]. Va gur beqref
fcrpvsvrq, gur genafyngvba ernqf
<display-eqn id="jstat256772eqn30" textype="equation" eqnnum="2.22" notation="TeX" lines="single" type="complex" font="italic" number="yes" numalign="bottom" eqnalign="indent"><?TeX \begin{equation}
\eqalign{
&\fl \{(2k,b),(1,3k)\}
\longleftrightarrow\{{\cal V}(\D_{2k,b}),{\cal V}(\D_{1,3k})\},\tqs
b=1,2,3;\quad k\in\mathbb{N}\\
&\fl \{(1,1),(1,2)\}
\longleftrightarrow\{\R^{(1)}(0)_{2},\R^{(1)}(0)_{1}\}\\
&\fl \{\R_{2,1}^{1,0},\R_{4,1}^{1,0},\R_{6,1}^{1,0},\R_{8,1}^{1,0}\}
\longleftrightarrow\{\R^{(2)}(0,2)_{7},\R^{(2)}(2,7),\R^{(2)}(7,15),\R^{(2)}(15,26)\}\\
&\fl
\{\R_{2,2}^{1,0},\R_{4,2}^{1,0},\R_{6,2}^{1,0},\R_{8,2}^{1,0}\}
\longleftrightarrow\{\R^{(2)}(0,1)_{5},\R^{(2)}(1,5),\R^{(2)}(5,12),\R^{(2)}(12,22)\}\\
&\fl
\{\R_{2,3}^{1,0},\R_{4,3}^{1,0},\R_{6,3}^{1,0},\R_{8,3}^{1,0}\}
\longleftrightarrow\{\R^{(2)}(1/3,1/3),\R^{(2)}(1/3,10/3),\R^{(2)}(10/3,28/3),\\
& \R^{(2)}(28/3,55/3)\}\\
& \fl
\{\R_{1,3}^{0,1},\R_{1,6}^{0,1},\R_{1,9}^{0,1},\R_{1,12}^{0,1}\}
\longleftrightarrow\{\R^{(2)}(0,1)_{7},\R^{(2)}(2,5),\R^{(2)}(7,12),\R^{(2)}(15,22)\}\\
& \fl
\{\R_{2,3}^{0,1},\R_{2,6}^{0,1},\R_{2,9}^{0,1},\R_{2,12}^{0,1}\}
\longleftrightarrow\{\R^{(2)}(1/8,1/8),\R^{(2)}(5/8,21/8),\R^{(2)}(33/8,65/8),\\
& \R^{(2)}(85/8,133/8)\}\\
& \fl
\{\R_{1,3}^{0,2},\R_{1,6}^{0,2},\R_{1,9}^{0,2},\R_{1,12}^{0,2}\}
\longleftrightarrow\{\R^{(2)}(0,2)_{5},\R^{(2)}(1,7),\R^{(2)}(5,15),\R^{(2)}(12,26)\}\\
& \fl
\{\R_{2,3}^{0,2},\R_{2,6}^{0,2},\R_{2,9}^{0,2},\R_{2,12}^{0,2}\}
\longleftrightarrow\{\R^{(2)}(5/8,5/8),\R^{(2)}(1/8,33/8),\R^{(2)}(21/8,85/8),\\
& \R^{(2)}(65/8,161/8)\}\\
& \fl
\{\R_{2,3}^{1,1},\R_{4,3}^{1,1},\R_{6,3}^{1,1},\R_{8,3}^{1,1}\}
\longleftrightarrow\{\R^{(3)}(0,0,1,1),\R^{(3)}(0,1,2,5),\R^{(3)}(2,5,7,12),\\
& \R^{(3)}(7,12,15,22)\}\\
& \fl
\{\R_{2,3}^{1,2},\R_{4,3}^{1,2},\R_{6,3}^{1,2},\R_{8,3}^{1,2}\}
\longleftrightarrow\{\R^{(3)}(0,0,2,2),\R^{(3)}(0,1,2,7),\R^{(3)}(1,5,7,15),\\
& \R^{(3)}(5,12,15,26)\}.}
\label{dictEF}
\end{equation} ?><eqn-graphic format="GIF" filename="images/jstat256772eqn30.gif" position="indent" align="middle"/></display-eqn>
Gur bayl guerr shfvbaf bs enax-6 vaqrpbzcbfnoyr ercerfragngvbaf pbafvqrerq va [<cite linkend="jstat256772bib45" show="author" override="no">50</cite>] pbeerfcbaq gb
<display-eqn id="jstat256772eqn31" textype="equation" eqnnum="2.23" notation="TeX" lines="single" type="complex" font="italic" number="yes" numalign="bottom" eqnalign="indent"><?TeX \begin{equation}
\eqalign{
\fl
\R_{2,3}^{1,1}\otimes \R_{2,3}^{1,1}=\R_{2,3}^{1,1}\oplus2\R_{4,3}^{1,1}\oplus
\R_{6,3}^{1,1}\oplus4\R_{2,3}^{1,2}\oplus2\R_{4,3}^{1,2}
\oplus8\R_{2,3}^{1,0}\oplus4\R_{4,3}^{1,0} \\
\fl
\R_{2,3}^{1,1}\otimes \R_{2,3}^{1,2}=4\R_{2,3}^{1,1}\oplus2\R_{4,3}^{1,1}\oplus
\R_{2,3}^{1,2}\oplus2\R_{4,3}^{1,2}\oplus\R_{6,3}^{1,2}\oplus2\R_{2,3}^{1,0}\oplus
4\R_{4,3}^{1,0}\oplus2\R_{6,3}^{1,0} \\
\fl
\R_{2,3}^{1,2}\otimes \R_{2,3}^{1,2}=\R_{2,3}^{1,1}\oplus2\R_{4,3}^{1,1}\oplus
\R_{6,3}^{1,1}\oplus4\R_{2,3}^{1,2}\oplus2\R_{4,3}^{1,2}\oplus2\R_{2,3}^{1,0}\\
\oplus\ 2\R_{4,3}^{1,0}\oplus2\R_{6,3}^{1,0}\oplus\R_{8,3}^{1,0}.}
\label{REF}
\end{equation} ?><eqn-graphic format="GIF" filename="images/jstat256772eqn31.gif" position="indent" align="middle"/></display-eqn>
Yvxrjvfr, gur bayl shfvba bs gur glcr (<eqnref linkend="jstat256772eqn24" override="no">7.81</eqnref>) pbafvqrerq va [<cite linkend="jstat256772bib45" show="author" override="no">50</cite>] pbeerfcbaqf gb
<display-eqn id="jstat256772eqn32" textype="equation" eqnnum="2.24" notation="TeX" lines="single" type="complex" font="italic" number="yes" numalign="bottom" eqnalign="indent"><?TeX \begin{equation}
\R_{2,2}^{1,0}\otimes \R_{2,3}^{0,2}=
2\R_{2,3}^{0,1}\oplus\R_{4,3}^{0,1}
\oplus(2,3)\oplus2(4,3)\oplus(6,3).
\end{equation} ?><eqn-graphic format="GIF" filename="images/jstat256772eqn32.gif" position="indent" align="middle"/></display-eqn>
Jr svaq gung bhe shfvba ehyrf erqhpr gb gur znal rknzcyrf rknzvarq ol Roreyr naq Sybue [<cite linkend="jstat256772bib45" show="author" override="no">50</cite>]. Guvf pbasvezf gurve bofreingvba gung vaqrpbzcbfnoyr ercerfragngvbaf bs
enax 6 ner erdhverq. Bhe erfhygf nyfb qrzbafgengr gung gur shfvba nytroen pybfrf
jvgubhg gur vagebqhpgvba bs vaqrpbzcbfnoyr ercerfragngvbaf bs uvture enax guna
6.</p>
        <p indent="yes" type="new-line">Roreyr naq Sybue nyfb cerfragrq na nytbevguz [<cite linkend="jstat256772bib45" show="author" override="no">50</cite>] sbe pbzchgvat shfvba cebqhpgf va gur nhtzragrq
<italic>p</italic><sub valign="no"><italic>c</italic>,<italic>c</italic>′</sub> zbqryf, va cnegvphyne
va gur nhtzragrq <italic>p</italic><sub valign="no">7,6</sub> = 9 
zbqry. Gurve nytbevguz vf ebbgrq va gur znal rkcyvpvg rknzcyrf rknzvarq va gurve cncre naq lvryqf
shfvba ehyrf juvpu ner obgu pbzzhgngvir naq nffbpvngvir. Pbafvqrevat gur nssvezngvir pbzcnevfba
bs bhe shfvba ehyrf jvgu gurve rknzcyrf, jr oryvrir gung gurve nytbevguz sbe gur nhtzragrq
<italic>p</italic><sub valign="no">7,6</sub> 
zbqry lvryqf erfhygf rdhvinyrag gb bhe rkcyvpvg shfvba ehyrf sbe pevgvpny crepbyngvba <inline-eqn notation="TeX" type="simple" font="italic"><?TeX {\cal
LM}(2,3) ?><inline-graphic format="GIF" filename="images/jstat256772ieqn4.gif" align="middle"/></inline-eqn>.
</p>
      </sec-level2>
      <sec-level2 id="jstat256772s2.7" label="2.7" appendix="no" type="num" style="arabic" inherit="yes">
        <heading>Xnp ercerfragngvbaf erivfvgrq</heading>
        <p indent="no" type="new-line">Nf nyernql vaqvpngrq naq nyfb qvfphffrq va [<cite linkend="jstat256772bib45" show="author" override="no">50</cite>], gur gjb ercerfragngvbaf
(8,8) 
naq (8,7) 
(gurer qrabgrq <inline-eqn notation="TeX" type="simple" font="italic"><?TeX \R^{(1)}(0)_{2} ?><inline-graphic format="GIF" filename="images/jstat256772ieqn38.gif" align="middle"/></inline-eqn> naq <inline-eqn notation="TeX" type="simple" font="italic"><?TeX \R^{(1)}(0)_{1} ?><inline-graphic format="GIF" filename="images/jstat256772ieqn39.gif" align="middle"/></inline-eqn>, erfcrpgviryl) ner abg shyyl erqhpvoyr. Jr dhbgr Roreyr naq Sybue:
<quote><p indent="yes" type="new-line">Ba gur bgure unaq, gur ercerfragngvbaf <inline-eqn notation="TeX" type="simple" font="italic"><?TeX \R^{(2)}(0,1)_5 ?><inline-graphic format="GIF" filename="images/jstat256772ieqn40.gif" align="middle"/></inline-eqn> naq <inline-eqn notation="TeX" type="simple" font="italic"><?TeX \R^{(2)}(0,1)_7 ?><inline-graphic format="GIF" filename="images/jstat256772ieqn41.gif" align="middle"/></inline-eqn> pbagnva n fgngr jvgu jrvtug 9 juvpu trarengrf n fhoercerfragngvba <inline-eqn notation="TeX" type="simple" font="italic"><?TeX \R^{(1)}(0)_1 ?><inline-graphic format="GIF" filename="images/jstat256772ieqn39.gif" align="middle"/></inline-eqn>. Guvf fhoercerfragngvba vf vaqrpbzcbfnoyr ohg arvgure vf vg veerqhpvoyr abe qbrf vg rkuvovg
nal uvture-enax orunivbhe. Vg bayl rkvfgf nf n fhoercerfragngvba nf vg arrqf gur rzorqqvat
vagb gur enax 7 ercerfragngvba va beqre abg gb unir ahyy-irpgbef ng obgu yriryf 8 naq 7. Ohg,
arireguryrff, orvat n fhoercerfragngvba bs n ercerfragngvba va gur fcrpgehz vg unf gb or
vapyhqrq vagb gur fcrpgehz, gbb.</p></quote>
Guvf vf pbeebobengrq ol bhe svaqvatf. Sebz gur ynggvpr, gur gjb ercerfragngvbaf
(8,8) 
naq (8,7) 
nevfr va gur pbasbezny fpnyvat yvzvg sebz irel fvzcyr naq angheny obhaqnel pbaqvgvbaf. Guvf
fhccbegf bhe nffregvba gung gurfr Xnp ercerfragngvbaf ner vaqrrq culfvpny. Shegurezber,
fvapr bar vf vzzrqvngryl snprq jvgu ceboyrzf jura nggrzcgvat gb vapyhqr gurve veerqhpvoyr
pbzcbaragf
<display-eqn id="jstat256772eqn33" textype="equation" eqnnum="2.25" notation="TeX" lines="single" type="complex" font="italic" number="yes" numalign="bottom" eqnalign="indent"><?TeX \begin{equation}
(1,1):\, \{{\cal V}(0),{\cal V}(2)\},\tqs
(1,2):\, \{{\cal V}(0),{\cal V}(1)\}
\end{equation} ?><eqn-graphic format="GIF" filename="images/jstat256772eqn33.gif" position="indent" align="middle"/></display-eqn>
va gur shfvba nytroen, jr nqibpngr gb pbafvqre shfvba nytroenf bs pevgvpny crepbyngvba trarengrq
sebz Xnp ercerfragngvbaf naq vaqrpbzcbfnoyr ercerfragngvbaf bs uvture enax. Gur bayl
veerqhpvoyr ercerfragngvbaf nccrnevat va gurfr shfvba nytroenf ner gurersber gurzfryirf Xnp
ercerfragngvbaf, gung vf, gurl orybat gb gur frg bs veerqhpvoyr Xnp ercerfragngvbaf
{(7<italic>x</italic>,8),(7<italic>x</italic>,7),(7<italic>x</italic>,6) = (7,6<italic>x</italic>),(8,6<italic>x</italic>)}. Angheny rkgrafvbaf bs gur ubevmbagny, iregvpny naq shaqnzragny shfvba nytroenf vaibyir <italic>nyy</italic>
gur nffbpvngrq Xnp ercerfragngvbaf naq ernq
<display-eqn id="jstat256772eqn34" textype="equation" eqnnum="2.26" notation="TeX" lines="single" type="complex" font="italic" number="yes" numalign="bottom" eqnalign="indent"><?TeX \begin{equation}
\fl
\bigl\langle(2,1),(3,1)\bigr\rangle,\tqs
\bigl\langle(1,2),(1,4)\bigr\rangle,\tqs
\bigl\langle(2,1),(3,1),(1,2),(1,4)\bigr\rangle
\label{full}
\end{equation} ?><eqn-graphic format="GIF" filename="images/jstat256772eqn34.gif" position="indent" align="middle"/></display-eqn>
erfcrpgviryl. Gurl jvyy or nqqerffrq ryfrjurer. Shegure rivqrapr va fhccbeg bs gur
eryrinapr bs Xnp ercerfragngvbaf va ybtnevguzvp PSG znl or sbhaq va [<cite linkend="jstat256772bib50" show="author" override="no">09</cite>]
jurer dhbgvrag zbqhyrf jvgu punenpgref (<eqnref linkend="jstat256772eqn2" override="no">8.7</eqnref>) ner sbhaq gb nevfr anghenyyl va gur yvzvg bs
pregnva frdhraprf bs zvavzny zbqryf.
</p>
      </sec-level2>
    </sec-level1>
    <sec-level1 id="jstat256772s3" label="3" appendix="no" type="num" style="arabic" inherit="yes">
      <heading>Pbapyhfvba</heading>
      <p indent="no" type="new-line">Jr unir cerfragrq rkcyvpvg trareny pbawrpgherf sbe gur puveny shfvba nytroenf bs pevgvpny
crepbyngvba, naq jr unir rkuvovgrq qvpgvbanevrf gb snpvyvgngr pbzcnevfba bs bhe erfhygf jvgu
gur cnegvphyne erfhygf bs Roreyr naq Sybue [<cite linkend="jstat256772bib45" show="author" override="no">50</cite>] naq Ernq naq Fnyrhe [<cite linkend="jstat256772bib48" show="author" override="no">53</cite>]. Vzcbegnagyl, jr bofreir gur nccrnenapr bs enax-6 vaqrpbzcbfnoyr
ercerfragngvbaf va nterrzrag jvgu Roreyr naq Sybue. Bhe shaqnzragny shfvba nytroen vf
ohvyg sebz vaqrcraqrag ubevmbagny naq iregvpny nytroenf gung, ng gur yriry bs punenpgref,
erfcrpg na haqreylvat <inline-eqn notation="TeX" type="simple" font="italic"><?TeX s\ell (2) ?><inline-graphic format="GIF" filename="images/jstat256772ieqn1.gif" align="middle"/></inline-eqn> fgehpgher. Gur vqragvgl (8,8) 
bs guvf shaqnzragny shfvba nytroen vf n erqhpvoyr lrg vaqrpbzcbfnoyr Xnp ercerfragngvba bs
enax 8.</p>
      <p indent="yes" type="new-line">Bhe ercbegrq shfvba ehyrf ner fhccbegrq ol rkgrafvir ahzrevpny vairfgvtngvbaf bs na
vagrtenoyr ynggvpr zbqry bs pevgvpny crepbyngvba. Gurfr ynggvpr erfhygf jvyy or cerfragrq
ryfrjurer. Jr nyfb ubcr gb qvfphff ryfrjurer gur shyy shfvba nytroen rapbzcnffvat nyy bs
gur Xnp ercerfragngvbaf nf jryy nf rkgrafvbaf gb trareny ybtnevguzvp zvavzny
zbqryf.</p>
    </sec-level1>
    <acknowledgment>
      <heading>Npxabjyrqtzragf</heading>
      <p indent="no" type="new-line">Guvf jbex jnf fhccbegrq ol gur Nhfgenyvna Erfrnepu Pbhapvy. WE gunaxf Naqernf Yhqjvt
sbe rapbhentvat qvfphffvbaf ng gur XVGC va Abirzore 7991.
</p>
    </acknowledgment>
  </body>
  <back>
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