4 1. SYMMETRI C FUNCTION S Pushing a box dow n give s a smaller partition , bu t i t is not tru e tha t i t gives a pai r o f consecutive partition s : an d ar e no t consecutive , becaus e D D D D D D I the mov e of the blac k bo x may b e performe d i n two steps : D D * , D D v n n n a n a * n a n * Let J , / b e a pair o f partitions suc h that th e diagram o f J contain s the diagra m of / . The n th e set difference o f the tw o diagram s i s called a skew diagram and denoted J/I (addin g commo n boxe s t o I an d J doe s no t change J/I. I n some problems, on e ha s to consider th e pair s (J,/ ) rathe r tha n J/I). If J/I contain s n o 2 x 2 sub-diagram an d i s connected (resp . J/I contain s no two boxe s i n the sam e column , res . n o two boxe s i n the sam e row) , the n J/I i s called a ribbon (resp . horizontal strip, resp . vertical strip). Ther e ar e strip s whic h are bot h vertica l an d horizontal , fo r exampl e a single box . :J••• DDDI DDDD DDDII DDDDI DDDDI DDDDD DDDDD DDDDI ribbon horizonta l stri p vertica l stri p A partitio n o f the typ e [1^ , a+1] i s called a hook an d i s denoted (a&/?) . The decomposition o f the diagra m o f a partition I int o it s diagonal hook s (i.e . hook s having thei r hea d o n th e diagonal ) i s called th e Frobenius code of / an d denote d $xob(I) (ai, #2? i&r &/3i,/?2, •, br) (wher e r , th e number o f boxes i n th e main diagonal , i s called th e rank o f the partition) . D / = [2,4,5,6 ] = g " 2 g B give s ffrob([2,4,5,6]) = (531 &320) . D D D D D D Given a box in a diagram, its content i s its distance to the mai n diagonal . Th e multiset o f contents permit s to recover th e diagram , henc e the partition. Replacin g each bo x b y its content, on e has , fo r example, tha t I = [2,4 , 5,6] ha s contents —3 —2 I 2 ~l °1 3 an d multise t {(-3) 1 , (-2) 2 , (-1) 2 , 0 3 , l 3 , 22, 3 2 , 41, 51}. 0 1 2 3 4 5 Finally (fo r th e moment!) , on e ma y cod e a partition b y th e wor d obtaine d by reading th e border of its diagra m : 0 for a n horizonta l step , 1 for a vertical step . 0 0 1 I = [2,4,5,6] = » borde r = D D D D 0 1 = [0,0,1,0,0,1,0,1,0,1 ] L ''' J D D D D D 0 1 L J D D D D D D Taking revers e word s an d exchangin g 0 and 1 corresponds to taking conjugat e partitions. I n fact, al l operations o n words o n two letter s induc e operation s on partitions tha t w e leave to the reade r th e pleasur e of discovering. It i s appropriate t o concatenate letter s 1 on the left o f a borde r word , and letters 0 on its right . Thi s amount s t o consider tha t th e partitio n b e containe d in a rectangula r partitio n m n (wher e m is the total numbe r o f O's an d n th e tota l number o f l's). Sinc e a 0,1 wor d i s specified b y it s lengt h an d th e positio n o f the O's, one ha s th e followin g lemma , whic h is needed i n determinantal identities . LEMMA 1.2.5 . Let I [i\,... ,i n ] E Nn be a partition contained in mn, and J = [ji,... , jm] be its conjugate. Then {zi+1, Z2+2 , ..., i n +n} and {m+n+ 1 ji 1, m+n+ 1 j2~2, ... , m+n+1 jm - m }
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