n-Örgü grubunun burav temsiline benzer temsilleri

Abstract

/>>=/> +i >(9) en+->=e>+->(10) g'/n+l>=g'n+l/n + ±>(11) &'n>=6'n + l>(12) 1n *^> s g/n+±>=gn+l/n + l>(13) d/n>=dn/n>(14) where n > and n + > belong to linear spaces V and W such that dimV - dimW and a,b,b',d:V->V f,g:W^V f',g':V-+W e:W^W. These definitions should satisfy the 17 equations arising from the relation (4). If öne applies (6-14) for 17 equations the relations among the elements of B are found as (l-O(l-O = (l-«M,)(l-<+1) = 3,(15) *n+en+dn^=2-q,(16) /n/: = (i-O(i-O,(17) gn+lg'n+l = (i - ^+1)(ı - O,(ıs) M'n = (l-<*n+1)(l-O,(19) f'nbngn+l = fnb'ng'n+l = («` - l)(en - 1)«+1 - 1).(20) Now, we try to find a representation in the case where B is unitary, i.e. BB^ = B^B = I where B^ is hennitean conjugate of B. Then (15-20) show that the Hecke algebra relation B2 = (l - q)B + q are satisfied. For unitary B, this relation linearizes as B = (l - q) + qB^. Using this we show that there is no ünite dimensional unitary representation. However for the pseudo unitary case where pl Gj3i = G finite dimensional representations exits and we construct such a representation with a diagonal G with Gz = I. Example. For a real T = TO, let an > l be chosen. Then ali en 's will be viü

/>>=/> + i > (9) en+->=e>+-> (10) g'/n+l>=g'n+l/n + ±> (11) b'/n>=b'n/n + l> (12) <7n+->=gn+1n + l> (13) d/n >= dn /n > (14) where /n > and /n + > belong to linear spaces V and W such that dimV = dimW and a,b,b',d:V->V f,g:W-+V f',g':V-+W These definitions should satisfy the 17 equations arising from the relation (4). If one applies (6-14) for 17 equations the relations among the elements of B axe found as (i-0(i-0 = (i-«.+x)(i-<U) = 3, (15) an+^+dn+1=2-q, (16) /`/I = (l-0(l-en), (17) gn+lg'n+l = (i - <*n+1)(i - O, (is) M>(i-<*.+i)(i-0, (19) = («. ` l)(e. - 1)«+1 - 1). (20) Now, we try to find a representation in the case where B is unitary, i.e. BB^ = B^B = I where B^ is hermitean conjugate of B. Then (15-20) show that the Hecke algebra relation B2 = (1 - q)B + q are satisfied. For unitary B, this relation linearizes as B - (1 - q) + qB^. Using this we show that there is no finite dimensional unitary representation. However for the pseudo unitary case where PjGfii = G finite dimensional representations exits and we construct such a representation with a diagonal G with G2 = I. Example. For a real r = r0, let an > 1 be chosen. Then all en 's will be viii/>>=/> +i >(9) en+->=e>+->(10) g'/n+l>=g'n+l/n + ±>(11) &'n>=6'n + l>(12) 1n *^> s g/n+±>=gn+l/n + l>(13) d/n>=dn/n>(14) where n > and n + > belong to linear spaces V and W such that dimV - dimW and a,b,b',d:V->V f,g:W^V f',g':V-+W e:W^W. These definitions should satisfy the 17 equations arising from the relation (4). If öne applies (6-14) for 17 equations the relations among the elements of B are found as (l-O(l-O = (l-«M,)(l-<+1) = 3,(15) *n+en+dn^=2-q,(16) /n/: = (i-O(i-O,(17) gn+lg'n+l = (i - ^+1)(ı - O,(ıs) M'n = (l-<*n+1)(l-O,(19) f'nbngn+l = fnb'ng'n+l = («` - l)(en - 1)«+1 - 1).(20) Now, we try to find a representation in the case where B is unitary, i.e. BB^ = B^B = I where B^ is hennitean conjugate of B. Then (15-20) show that the Hecke algebra relation B2 = (l - q)B + q are satisfied. For unitary B, this relation linearizes as B = (l - q) + qB^. Using this we show that there is no ünite dimensional unitary representation. However for the pseudo unitary case where pl Gj3i = G finite dimensional representations exits and we construct such a representation with a diagonal G with Gz = I. Example. For a real T = TO, let an > l be chosen. Then ali en 's will be viü/>>=/> + i > (9) en+->=e>+-> (10) g'/n+l>=g'n+l/n + ±> (11) b'/n>=b'n/n + l> (12) <7n+->=gn+1n + l> (13) d/n >= dn /n > (14) where /n > and /n + > belong to linear spaces V and W such that dimV = dimW and a,b,b',d:V->V f,g:W-+V f',g':V-+W These definitions should satisfy the 17 equations arising from the relation (4). If one applies (6-14) for 17 equations the relations among the elements of B axe found as (i-0(i-0 = (i-«.+x)(i-<U) = 3, (15) an+^+dn+1=2-q, (16) /`/I = (l-0(l-en), (17) gn+lg'n+l = (i - <*n+1)(i - O, (is) M>(i-<*.+i)(i-0, (19) = («. ` l)(e. - 1)«+1 - 1). (20) Now, we try to find a representation in the case where B is unitary, i.e. BB^ = B^B = I where B^ is hermitean conjugate of B. Then (15-20) show that the Hecke algebra relation B2 = (1 - q)B + q are satisfied. For unitary B, this relation linearizes as B - (1 - q) + qB^. Using this we show that there is no finite dimensional unitary representation. However for the pseudo unitary case where PjGfii = G finite dimensional representations exits and we construct such a representation with a diagonal G with G2 = I. Example. For a real r = r0, let an > 1 be chosen. Then all en 's will be viii/>>=/> +i >(9) en+->=e>+->(10) g'/n+l>=g'n+l/n + ±>(11) &'n>=6'n + l>(12) 1n *^> s g/n+±>=gn+l/n + l>(13) d/n>=dn/n>(14) where n > and n + > belong to linear spaces V and W such that dimV - dimW and a,b,b',d:V->V f,g:W^V f',g':V-+W e:W^W. These definitions should satisfy the 17 equations arising from the relation (4). If öne applies (6-14) for 17 equations the relations among the elements of B are found as (l-O(l-O = (l-«M,)(l-<+1) = 3,(15) *n+en+dn^=2-q,(16) /n/: = (i-O(i-O,(17) gn+lg'n+l = (i - ^+1)(ı - O,(ıs) M'n = (l-<*n+1)(l-O,(19) f'nbngn+l = fnb'ng'n+l = («` - l)(en - 1)«+1 - 1).(20) Now, we try to find a representation in the case where B is unitary, i.e. BB^ = B^B = I where B^ is hennitean conjugate of B. Then (15-20) show that the Hecke algebra relation B2 = (l - q)B + q are satisfied. For unitary B, this relation linearizes as B = (l - q) + qB^. Using this we show that there is no ünite dimensional unitary representation. However for the pseudo unitary case where pl Gj3i = G finite dimensional representations exits and we construct such a representation with a diagonal G with Gz = I. Example. For a real T = TO, let an > l be chosen. Then ali en 's will be viü/>>=/> + i > (9) en+->=e>+-> (10) g'/n+l>=g'n+l/n + ±> (11) b'/n>=b'n/n + l> (12) <7n+->=gn+1n + l> (13) d/n >= dn /n > (14) where /n > and /n + > belong to linear spaces V and W such that dimV = dimW and a,b,b',d:V->V f,g:W-+V f',g':V-+W These definitions should satisfy the 17 equations arising from the relation (4). If one applies (6-14) for 17 equations the relations among the elements of B axe found as (i-0(i-0 = (i-«.+x)(i-<U) = 3, (15) an+^+dn+1=2-q, (16) /`/I = (l-0(l-en), (17) gn+lg'n+l = (i - <*n+1)(i - O, (is) M>(i-<*.+i)(i-0, (19) = («. ` l)(e. - 1)«+1 - 1). (20) Now, we try to find a representation in the case where B is unitary, i.e. BB^ = B^B = I where B^ is hermitean conjugate of B. Then (15-20) show that the Hecke algebra relation B2 = (1 - q)B + q are satisfied. For unitary B, this relation linearizes as B - (1 - q) + qB^. Using this we show that there is no finite dimensional unitary representation. However for the pseudo unitary case where PjGfii = G finite dimensional representations exits and we construct such a representation with a diagonal G with G2 = I. Example. For a real r = r0, let an > 1 be chosen. Then all en 's will be viii