Sums of triangular numbers from the Frobenius determinant
- 作者：Hjalmar Rosengren
- 关键词：Sums of triangular numbers ; Pfaffian ; Frobenius determinant ; Denominator formula ; Affine superalgebra
- 刊名：Advances in Mathematics
- 出版年：2007
- 期刊代码：62_00018708
- 类别：mt
- 出版时间：30 January 2007
- 卷：208
- 期：2
- 页码：935-961
- 文件大小：235 K
摘要
We show that the denominator formula for the strange series of affine superalgebras, conjectured by Kac and Wakimoto and proved by Zagier, follows from a classical determinant evaluation of Frobenius. As a limit case, we obtain exact formulas for the number of representations of an arbitrary number as a sum of decoration:none; color:black" href="/science?_ob=MathURL&_method=retrieve&_udi=B6W9F-4K2SKB3-1&_mathId=mml1&_user=10&_cdi=6681&_rdoc=19&_acct=C000050221&_version=1&_userid=10&md5=68df1761991770330f85a8c8319e73ec" title="Click to view the MathML source">4m2/d triangles, whenever decoration:none; color:black" href="/science?_ob=MathURL&_method=retrieve&_udi=B6W9F-4K2SKB3-1&_mathId=mml2&_user=10&_cdi=6681&_rdoc=19&_acct=C000050221&_version=1&_userid=10&md5=2685111f61e9a877129baae39155f83e" title="Click to view the MathML source">d|2m, and decoration:none; color:black" href="/science?_ob=MathURL&_method=retrieve&_udi=B6W9F-4K2SKB3-1&_mathId=mml3&_user=10&_cdi=6681&_rdoc=19&_acct=C000050221&_version=1&_userid=10&md5=5820810b0bf68a31e0060e1501c6415e" title="Click to view the MathML source">4m(m+1)/d triangles, when decoration:none; color:black" href="/science?_ob=MathURL&_method=retrieve&_udi=B6W9F-4K2SKB3-1&_mathId=mml4&_user=10&_cdi=6681&_rdoc=19&_acct=C000050221&_version=1&_userid=10&md5=44c19cbbd1352da5757d6e9f5a91e485" title="Click to view the MathML source">d|2m or decoration:none; color:black" href="/science?_ob=MathURL&_method=retrieve&_udi=B6W9F-4K2SKB3-1&_mathId=mml5&_user=10&_cdi=6681&_rdoc=19&_acct=C000050221&_version=1&_userid=10&md5=a864b3eb7a3f8c65625fa93ffad5238d" title="Click to view the MathML source">d|2m+2. This extends recent results of Getz and Mahlburg, Milne, and Zagier.