On the commutative quotient of Fomin-Kirillov algebras

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• 作者：Ricky Ini Liu ^{riliu@ncsu.edu" class="auth_mail" title="E-mail the corresponding author}
• 刊名：European Journal of Combinatorics
• 出版年：2016
• 出版时间：May 2016
• 年：2016
• 卷：54
• 期：Complete
• 页码：65-75
• 全文大小：387 K

文摘

The Fomin–Kirillov algebra an id="mmlsi3" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S019566981500267X&_mathId=si3.gif&_user=111111111&_pii=S019566981500267X&_rdoc=1&_issn=01956698&md5=d36588b44a5737ba7464c8b79595f21e" title="Click to view the MathML source">E_nan>an class="mathContainer hidden">an class="mathCode">ath altimg="si3.gif" overflow="scroll">athvariant="script">Enath>an>an>an> is a noncommutative algebra with a generator for each edge of the complete graph on an id="mmlsi4" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S019566981500267X&_mathId=si4.gif&_user=111111111&_pii=S019566981500267X&_rdoc=1&_issn=01956698&md5=9c36bed3a3f029bc88fa9c2f9dd23acd" title="Click to view the MathML source">nan>an class="mathContainer hidden">an class="mathCode">ath altimg="si4.gif" overflow="scroll">nath>an>an>an> vertices. For any graph an id="mmlsi5" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S019566981500267X&_mathId=si5.gif&_user=111111111&_pii=S019566981500267X&_rdoc=1&_issn=01956698&md5=4bd1bb0dccaf1981c119ee13b7795113" title="Click to view the MathML source">Gan>an class="mathContainer hidden">an class="mathCode">ath altimg="si5.gif" overflow="scroll">Gath>an>an>an> on an id="mmlsi4" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S019566981500267X&_mathId=si4.gif&_user=111111111&_pii=S019566981500267X&_rdoc=1&_issn=01956698&md5=9c36bed3a3f029bc88fa9c2f9dd23acd" title="Click to view the MathML source">nan>an class="mathContainer hidden">an class="mathCode">ath altimg="si4.gif" overflow="scroll">nath>an>an>an> vertices, let an id="mmlsi7" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S019566981500267X&_mathId=si7.gif&_user=111111111&_pii=S019566981500267X&_rdoc=1&_issn=01956698&md5=ce3811ff02539e87a40a4fcc206f4861" title="Click to view the MathML source">E_Gan>an class="mathContainer hidden">an class="mathCode">ath altimg="si7.gif" overflow="scroll">athvariant="script">EGath>an>an>an> be the subalgebra of an id="mmlsi3" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S019566981500267X&_mathId=si3.gif&_user=111111111&_pii=S019566981500267X&_rdoc=1&_issn=01956698&md5=d36588b44a5737ba7464c8b79595f21e" title="Click to view the MathML source">E_nan>an class="mathContainer hidden">an class="mathCode">ath altimg="si3.gif" overflow="scroll">athvariant="script">Enath>an>an>an> generated by the edges in an id="mmlsi5" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S019566981500267X&_mathId=si5.gif&_user=111111111&_pii=S019566981500267X&_rdoc=1&_issn=01956698&md5=4bd1bb0dccaf1981c119ee13b7795113" title="Click to view the MathML source">Gan>an class="mathContainer hidden">an class="mathCode">ath altimg="si5.gif" overflow="scroll">Gath>an>an>an>. We show that the commutative quotient of an id="mmlsi7" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S019566981500267X&_mathId=si7.gif&_user=111111111&_pii=S019566981500267X&_rdoc=1&_issn=01956698&md5=ce3811ff02539e87a40a4fcc206f4861" title="Click to view the MathML source">E_Gan>an class="mathContainer hidden">an class="mathCode">ath altimg="si7.gif" overflow="scroll">athvariant="script">EGath>an>an>an> is isomorphic to the Orlik–Terao algebra of an id="mmlsi5" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S019566981500267X&_mathId=si5.gif&_user=111111111&_pii=S019566981500267X&_rdoc=1&_issn=01956698&md5=4bd1bb0dccaf1981c119ee13b7795113" title="Click to view the MathML source">Gan>an class="mathContainer hidden">an class="mathCode">ath altimg="si5.gif" overflow="scroll">Gath>an>an>an>. As a consequence, the Hilbert series of this quotient is given by an id="mmlsi12" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S019566981500267X&_mathId=si12.gif&_user=111111111&_pii=S019566981500267X&_rdoc=1&_issn=01956698&md5=679b31357ac147578819eab780270841" title="Click to view the MathML source">(−t)ⁿχ_G(−t⁻¹)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si12.gif" overflow="scroll">(−t)nχG(−t−1)ath>an>an>an>, where an id="mmlsi13" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S019566981500267X&_mathId=si13.gif&_user=111111111&_pii=S019566981500267X&_rdoc=1&_issn=01956698&md5=20c8c5c596840bf7eb525ef37b3f7c97" title="Click to view the MathML source">χ_Gan>an class="mathContainer hidden">an class="mathCode">ath altimg="si13.gif" overflow="scroll">χGath>an>an>an> is the chromatic polynomial of an id="mmlsi5" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S019566981500267X&_mathId=si5.gif&_user=111111111&_pii=S019566981500267X&_rdoc=1&_issn=01956698&md5=4bd1bb0dccaf1981c119ee13b7795113" title="Click to view the MathML source">Gan>an class="mathContainer hidden">an class="mathCode">ath altimg="si5.gif" overflow="scroll">Gath>an>an>an>. We also give a reduction algorithm for the graded components of an id="mmlsi7" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S019566981500267X&_mathId=si7.gif&_user=111111111&_pii=S019566981500267X&_rdoc=1&_issn=01956698&md5=ce3811ff02539e87a40a4fcc206f4861" title="Click to view the MathML source">E_Gan>an class="mathContainer hidden">an class="mathCode">ath altimg="si7.gif" overflow="scroll">athvariant="script">EGath>an>an>an> that do not vanish in the commutative quotient and show that their structure is described by the combinatorics of noncrossing forests.