Pub Date : 2022-03-09DOI: 10.1017/s1446788722000179
Nick Bridger, Masoud Kamgarpour
We compute the number of points over finite fields of the character stack associated to a compact surface group and a reductive group with connected centre. We find that the answer is a polynomial on residue classes (PORC). The key ingredients in the proof are Lusztig’s Jordan decomposition of complex characters of finite reductive groups and Deriziotis’s results on their genus numbers. As a consequence of our main theorem, we obtain an expression for the E-polynomial of the character stack.
{"title":"CHARACTER STACKS ARE PORC COUNT","authors":"Nick Bridger, Masoud Kamgarpour","doi":"10.1017/s1446788722000179","DOIUrl":"https://doi.org/10.1017/s1446788722000179","url":null,"abstract":"\u0000 We compute the number of points over finite fields of the character stack associated to a compact surface group and a reductive group with connected centre. We find that the answer is a polynomial on residue classes (PORC). The key ingredients in the proof are Lusztig’s Jordan decomposition of complex characters of finite reductive groups and Deriziotis’s results on their genus numbers. As a consequence of our main theorem, we obtain an expression for the E-polynomial of the character stack.","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"12 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75454684","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-03-08DOI: 10.1017/S1446788721000392
V. Deaconu
Abstract We introduce certain $C^*$ -algebras and k-graphs associated to k finite-dimensional unitary representations $rho _1,ldots ,rho _k$ of a compact group G. We define a higher rank Doplicher-Roberts algebra $mathcal {O}_{rho _1,ldots ,rho _k}$ , constructed from intertwiners of tensor powers of these representations. Under certain conditions, we show that this $C^*$ -algebra is isomorphic to a corner in the $C^*$ -algebra of a row-finite rank k graph $Lambda $ with no sources. For G finite and $rho _i$ faithful of dimension at least two, this graph is irreducible, it has vertices $hat {G}$ and the edges are determined by k commuting matrices obtained from the character table of the group. We illustrate this with some examples when $mathcal {O}_{rho _1,ldots ,rho _k}$ is simple and purely infinite, and with some K-theory computations.
{"title":"$boldsymbol {C}^{*}$\u0000 -ALGEBRAS FROM \u0000$boldsymbol {K}$\u0000 GROUP REPRESENTATIONS","authors":"V. Deaconu","doi":"10.1017/S1446788721000392","DOIUrl":"https://doi.org/10.1017/S1446788721000392","url":null,"abstract":"Abstract We introduce certain \u0000$C^*$\u0000 -algebras and k-graphs associated to k finite-dimensional unitary representations \u0000$rho _1,ldots ,rho _k$\u0000 of a compact group G. We define a higher rank Doplicher-Roberts algebra \u0000$mathcal {O}_{rho _1,ldots ,rho _k}$\u0000 , constructed from intertwiners of tensor powers of these representations. Under certain conditions, we show that this \u0000$C^*$\u0000 -algebra is isomorphic to a corner in the \u0000$C^*$\u0000 -algebra of a row-finite rank k graph \u0000$Lambda $\u0000 with no sources. For G finite and \u0000$rho _i$\u0000 faithful of dimension at least two, this graph is irreducible, it has vertices \u0000$hat {G}$\u0000 and the edges are determined by k commuting matrices obtained from the character table of the group. We illustrate this with some examples when \u0000$mathcal {O}_{rho _1,ldots ,rho _k}$\u0000 is simple and purely infinite, and with some K-theory computations.","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"34 1","pages":"318 - 338"},"PeriodicalIF":0.7,"publicationDate":"2022-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90988189","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-03-07DOI: 10.1017/s1446788721000276
{"title":"JAZ volume 112 issue 2 Cover and Back matter","authors":"","doi":"10.1017/s1446788721000276","DOIUrl":"https://doi.org/10.1017/s1446788721000276","url":null,"abstract":"","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"27 1","pages":"b1 - b2"},"PeriodicalIF":0.7,"publicationDate":"2022-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89916172","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-03-07DOI: 10.1017/s1446788721000264
{"title":"JAZ volume 112 issue 2 Cover and Front matter","authors":"","doi":"10.1017/s1446788721000264","DOIUrl":"https://doi.org/10.1017/s1446788721000264","url":null,"abstract":"","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"12 1","pages":"f1 - f2"},"PeriodicalIF":0.7,"publicationDate":"2022-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74721989","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-26DOI: 10.1017/S1446788721000409
W. Jin, L. Tan
Abstract A noncomplete graph is $2$ -distance-transitive if, for $i in {1,2}$ and for any two vertex pairs $(u_1,v_1)$ and $(u_2,v_2)$ with the same distance i in the graph, there exists an element of the graph automorphism group that maps $(u_1,v_1)$ to $(u_2,v_2)$ . This paper determines the family of $2$ -distance-transitive Cayley graphs over dihedral groups, and it is shown that if the girth of such a graph is not $4$ , then either it is a known $2$ -arc-transitive graph or it is isomorphic to one of the following two graphs: $ {mathrm {K}}_{x[y]}$ , where $xgeq 3,ygeq 2$ , and $G(2,p,({p-1})/{4})$ , where p is a prime and $p equiv 1 (operatorname {mod}, 8)$ . Then, as an application of the above result, a complete classification is achieved of the family of $2$ -geodesic-transitive Cayley graphs for dihedral groups.
{"title":"FINITE TWO-DISTANCE-TRANSITIVE DIHEDRANTS","authors":"W. Jin, L. Tan","doi":"10.1017/S1446788721000409","DOIUrl":"https://doi.org/10.1017/S1446788721000409","url":null,"abstract":"Abstract A noncomplete graph is \u0000$2$\u0000 -distance-transitive if, for \u0000$i in {1,2}$\u0000 and for any two vertex pairs \u0000$(u_1,v_1)$\u0000 and \u0000$(u_2,v_2)$\u0000 with the same distance i in the graph, there exists an element of the graph automorphism group that maps \u0000$(u_1,v_1)$\u0000 to \u0000$(u_2,v_2)$\u0000 . This paper determines the family of \u0000$2$\u0000 -distance-transitive Cayley graphs over dihedral groups, and it is shown that if the girth of such a graph is not \u0000$4$\u0000 , then either it is a known \u0000$2$\u0000 -arc-transitive graph or it is isomorphic to one of the following two graphs: \u0000$ {mathrm {K}}_{x[y]}$\u0000 , where \u0000$xgeq 3,ygeq 2$\u0000 , and \u0000$G(2,p,({p-1})/{4})$\u0000 , where p is a prime and \u0000$p equiv 1 (operatorname {mod}, 8)$\u0000 . Then, as an application of the above result, a complete classification is achieved of the family of \u0000$2$\u0000 -geodesic-transitive Cayley graphs for dihedral groups.","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"56 1","pages":"386 - 401"},"PeriodicalIF":0.7,"publicationDate":"2022-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78424128","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-21DOI: 10.1017/S1446788721000239
Chih-Whi Chen, V. Mazorchuk
Abstract For classical Lie superalgebras of type I, we provide necessary and sufficient conditions for a Verma supermodule $Delta (lambda )$ to be such that every nonzero homomorphism from another Verma supermodule to $Delta (lambda )$ is injective. This is applied to describe the socle of the cokernel of an inclusion of Verma supermodules over the periplectic Lie superalgebras $mathfrak {pe} (n)$ and, furthermore, to reduce the problem of description of $mathrm {Ext}^1_{mathcal O}(L(mu ),Delta (lambda ))$ for $mathfrak {pe} (n)$ to the similar problem for the Lie algebra $mathfrak {gl}(n)$ . Additionally, we study the projective and injective dimensions of structural supermodules in parabolic category $mathcal O^{mathfrak {p}}$ for classical Lie superalgebras. In particular, we completely determine these dimensions for structural supermodules over the periplectic Lie superalgebra $mathfrak {pe} (n)$ and the orthosymplectic Lie superalgebra $mathfrak {osp}(2|2n)$ .
{"title":"SOME HOMOLOGICAL PROPERTIES OF CATEGORY $boldsymbol {mathcal {O}}$ FOR LIE SUPERALGEBRAS","authors":"Chih-Whi Chen, V. Mazorchuk","doi":"10.1017/S1446788721000239","DOIUrl":"https://doi.org/10.1017/S1446788721000239","url":null,"abstract":"Abstract For classical Lie superalgebras of type I, we provide necessary and sufficient conditions for a Verma supermodule \u0000$Delta (lambda )$\u0000 to be such that every nonzero homomorphism from another Verma supermodule to \u0000$Delta (lambda )$\u0000 is injective. This is applied to describe the socle of the cokernel of an inclusion of Verma supermodules over the periplectic Lie superalgebras \u0000$mathfrak {pe} (n)$\u0000 and, furthermore, to reduce the problem of description of \u0000$mathrm {Ext}^1_{mathcal O}(L(mu ),Delta (lambda ))$\u0000 for \u0000$mathfrak {pe} (n)$\u0000 to the similar problem for the Lie algebra \u0000$mathfrak {gl}(n)$\u0000 . Additionally, we study the projective and injective dimensions of structural supermodules in parabolic category \u0000$mathcal O^{mathfrak {p}}$\u0000 for classical Lie superalgebras. In particular, we completely determine these dimensions for structural supermodules over the periplectic Lie superalgebra \u0000$mathfrak {pe} (n)$\u0000 and the orthosymplectic Lie superalgebra \u0000$mathfrak {osp}(2|2n)$\u0000 .","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"75 1","pages":"50 - 77"},"PeriodicalIF":0.7,"publicationDate":"2022-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86168445","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-11DOI: 10.1017/s1446788721000240
{"title":"JAZ volume 112 issue 1 Cover and Front matter","authors":"","doi":"10.1017/s1446788721000240","DOIUrl":"https://doi.org/10.1017/s1446788721000240","url":null,"abstract":"","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"8 1","pages":"f1 - f2"},"PeriodicalIF":0.7,"publicationDate":"2022-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84289513","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-11DOI: 10.1017/s1446788721000252
{"title":"JAZ volume 112 issue 1 Cover and Back matter","authors":"","doi":"10.1017/s1446788721000252","DOIUrl":"https://doi.org/10.1017/s1446788721000252","url":null,"abstract":"","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"222 1","pages":"b1 - b2"},"PeriodicalIF":0.7,"publicationDate":"2022-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72701025","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-12-23DOI: 10.1017/S1446788722000180
C'edric Bonnaf'e
Abstract We study the symplectic leaves of the subvariety of fixed points of an automorphism of a Calogero–Moser space induced by an element of finite order of the normalizer of the associated complex reflection group. We give a parametrization à la Harish-Chandra of its symplectic leaves (generalizing earlier works of Bellamy and Losev). This result is inspired by the mysterious relations between the geometry of Calogero–Moser spaces and unipotent representations of finite reductive groups, which is the theme of another paper, C. Bonnafé [‘Calogero–Moser spaces vs unipotent representations’, Pure Appl. Math. Q., to appear, Preprint, 2021, arXiv:2112.13684].
研究了由相关复反射群的正则化项的有限阶元素所导出的Calogero-Moser空间的自同构不动点的子簇的辛叶。我们给出了它的辛叶的参数化(概括了Bellamy和Losev的早期工作)。这个结果的灵感来自于Calogero-Moser空间的几何与有限约化群的单幂表示之间的神秘关系，这是另一篇论文的主题，C. bonnaf [' Calogero-Moser空间与单幂表示'，纯应用。数学。[j].， to appear，预印本，2021,arXiv:2112.13684]。
{"title":"AUTOMORPHISMS AND SYMPLECTIC LEAVES OF CALOGERO–MOSER SPACES","authors":"C'edric Bonnaf'e","doi":"10.1017/S1446788722000180","DOIUrl":"https://doi.org/10.1017/S1446788722000180","url":null,"abstract":"Abstract We study the symplectic leaves of the subvariety of fixed points of an automorphism of a Calogero–Moser space induced by an element of finite order of the normalizer of the associated complex reflection group. We give a parametrization à la Harish-Chandra of its symplectic leaves (generalizing earlier works of Bellamy and Losev). This result is inspired by the mysterious relations between the geometry of Calogero–Moser spaces and unipotent representations of finite reductive groups, which is the theme of another paper, C. Bonnafé [‘Calogero–Moser spaces vs unipotent representations’, Pure Appl. Math. Q., to appear, Preprint, 2021, arXiv:2112.13684].","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"93 1","pages":"26 - 57"},"PeriodicalIF":0.7,"publicationDate":"2021-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80269795","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-23DOI: 10.1017/S1446788721000203
S. Pitchai Murugan, G. P. Youvaraj
Abstract Gabardo and Nashed [‘Nonuniform multiresolution analyses and spectral pairs’, J. Funct. Anal. 158(1) (1998), 209–241] have introduced the concept of nonuniform multiresolution analysis (NUMRA), based on the theory of spectral pairs, in which the associated translated set $Lambda ={0,{r}/{N}}+2mathbb Z$ is not necessarily a discrete subgroup of $mathbb{R}$ , and the translation factor is $2textrm{N}$ . Here r is an odd integer with $1leq rleq 2N-1$ such that r and N are relatively prime. The nonuniform wavelets associated with NUMRA can be used in signal processing, sampling theory, speech recognition and various other areas, where instead of integer shifts nonuniform shifts are needed. In order to further generalize this useful NUMRA, we consider the set $widetilde {Lambda }={0,{r_1}/{N},{r_2}/{N},ldots ,{r_q}/{N}}+smathbb Z$ , where s is an even integer, $qin mathbb {N}$ , $r_i$ is an integer such that $1leq r_ileq sN-1,,(r_i,N)=1$ for all i and $Ngeq 2$ . In this paper, we prove that the concept of NUMRA with the translation set $widetilde {Lambda }$ is possible only if $widetilde {Lambda }$ is of the form ${0,{r}/{N}}+smathbb Z$ . Next we introduce $Lambda _s$ -nonuniform multiresolution analysis ( $Lambda _s$ -NUMRA) for which the translation set is $Lambda _s={0,{r}/{N}}+smathbb Z$ and the dilation factor is $sN$ , where s is an even integer. Also, we characterize the scaling functions associated with $Lambda _s$ -NUMRA and we give necessary and sufficient conditions for wavelet filters associated with $Lambda _s$ -NUMRA.
{"title":"$Lambda _s$\u0000 -NONUNIFORM MULTIRESOLUTION ANALYSIS","authors":"S. Pitchai Murugan, G. P. Youvaraj","doi":"10.1017/S1446788721000203","DOIUrl":"https://doi.org/10.1017/S1446788721000203","url":null,"abstract":"Abstract Gabardo and Nashed [‘Nonuniform multiresolution analyses and spectral pairs’, J. Funct. Anal. 158(1) (1998), 209–241] have introduced the concept of nonuniform multiresolution analysis (NUMRA), based on the theory of spectral pairs, in which the associated translated set \u0000$Lambda ={0,{r}/{N}}+2mathbb Z$\u0000 is not necessarily a discrete subgroup of \u0000$mathbb{R}$\u0000 , and the translation factor is \u0000$2textrm{N}$\u0000 . Here r is an odd integer with \u0000$1leq rleq 2N-1$\u0000 such that r and N are relatively prime. The nonuniform wavelets associated with NUMRA can be used in signal processing, sampling theory, speech recognition and various other areas, where instead of integer shifts nonuniform shifts are needed. In order to further generalize this useful NUMRA, we consider the set \u0000$widetilde {Lambda }={0,{r_1}/{N},{r_2}/{N},ldots ,{r_q}/{N}}+smathbb Z$\u0000 , where s is an even integer, \u0000$qin mathbb {N}$\u0000 , \u0000$r_i$\u0000 is an integer such that \u0000$1leq r_ileq sN-1,,(r_i,N)=1$\u0000 for all i and \u0000$Ngeq 2$\u0000 . In this paper, we prove that the concept of NUMRA with the translation set \u0000$widetilde {Lambda }$\u0000 is possible only if \u0000$widetilde {Lambda }$\u0000 is of the form \u0000${0,{r}/{N}}+smathbb Z$\u0000 . Next we introduce \u0000$Lambda _s$\u0000 -nonuniform multiresolution analysis ( \u0000$Lambda _s$\u0000 -NUMRA) for which the translation set is \u0000$Lambda _s={0,{r}/{N}}+smathbb Z$\u0000 and the dilation factor is \u0000$sN$\u0000 , where s is an even integer. Also, we characterize the scaling functions associated with \u0000$Lambda _s$\u0000 -NUMRA and we give necessary and sufficient conditions for wavelet filters associated with \u0000$Lambda _s$\u0000 -NUMRA.","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"1 1","pages":"359 - 377"},"PeriodicalIF":0.7,"publicationDate":"2021-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82633940","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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