Amrican Journal of Mathmatics and Statistics 014, 4(): 58-64 DOI: 10593/jajms0140400 (Upsid-Down o Dirct Rotation) β - Numbrs Ammar Sddiq Mahmood 1, Shukriyah Sabir Ali,* 1 Dpartmnt of Mathmatics, Collg of Education, Univrsity of Mosul, Iraq Dpartmnt of Mathmatics, Collg of Computrs Scincs and Mathmatics, Univrsity of Mosul, Iraq Abstract For any partition μ = ( μ 1, μ, μ n ) of a non - ngativ intgr numbr r thr xist a diagram (A) of β - numbrs for ach whr is a positiv intgr numbr gratr than or qual to two; which introducd by Jams in 1978 Ths diagrams (A) play an normous rol in Iwahori-Hck algbras and q-schur algbras; as prsntd by Fayrs in 007 Mahmood gav nw diagrams by applying th upsid- down application on th main diagram (A) in 013 Anothr nw diagrams wr prsntd by th authors by applying th dirct rotation application on th main diagram (A) in 013 In th prsnt papr, w introducd som othr nw diagrams (A 1 ), (A ) and (A 3 ) by mploying th "composition of upsid- down application with dirct rotation application of thr diffrnt dgrs namly 90 o, 180 o and 70 o rspctivly on th main diagram (A) W concludd that w can find th succssiv main diagrams (A 1 ), (A ) and (A 3 ) for th guids b, b 3, and b dpnding on th main diagrams (A 1 ), (A ) and (A 3 ) for b 1 and st ths facts as ruls namd Rul (31), Rul (3) and Rul () rspctivly Kywords β - numbrs, Diagram (A), Dirct Rotation, Intrsction, Partition, Upsid-Down 1 Introduction Partition thory has a pivotal impact on numbr thory and has in addition an applid impact on rprsntation thory which is on of th most important lmnts of modrn algbra This is rprsntd by many studis on this topic, for xampl [1,3,4] Th prsnt papr dals basically with th subjct of rprsntation thory whr Young diagram plays an important rol in th drafting of th first stp of many typs of algbras What bnfits us hr is (lwahori - Hck algbras and q-schur algbras) Jams (1978), put th nw vrsion instad of Young's spcial diagram of spcific composition of positiv intgr numbrs which th sum of thm is a non ngativ intgr numbr calld r H noticd that th nw diagram won't work unlss th composition is a partition which satisfis th condition (µ i µ i+1, i) and h calld this, Diagram (A) Thn h continud putting a nw condition whn h said thr xists, whr is an intgr numbr gratr than or qual to It is according to this numbr that w will divid th runnrs of diagram (A) Initially was takn as a prim intgr numbr, so th rsults wr spcific Fayrs (007), abolishd th condition on bing a prim numbr Accordingly, th rsults wr too many to giv othrs nw scintific capabilitis This subjct has a connction with rprsntation thory of lwahori- Hck algbras and q-schur algbras [4] An xcllnt introduction to th rprsntation thory of Iwahori Hck * Corrsponding author: ssahiq@yahoocom (Shukriyah Sabir Ali) Publishd onlin at http://journalsapuborg/ajms Copyright 014 Scintific & Acadmic Publishing All Rights Rsrvd algbras and q-schur algbras can b found in [3], which also contains th dfinition of intgr partition Mor dtail on th lattr and β - numbrs was givn by Jams (1978) Counting th β - numbrs for any partition μ of r rquirs th dfinition of an intgr b which was showd latr by Mohammad (008), that it must b gratr than or qual to th numbr of parts of μ A S Mahmood (011), introducd th dfinition of main diagram (s) (A) and th ida of thir intrsction S M Mahmood (011), concludd that th convrsion of any partition μ of r to diagram (A) of β-numbrs maks it asy to idntify many proprtis inhrnt in th partition much mor than putting it in Young diagrams as boxs adjacnt to ach othr Mahmood (013), gav nw diagrams by applying th upsid-down application on th main diagram (A) Othr nw diagrams wr prsntd by th authors (013), by applying th dirct rotation application on th main diagram (A) In th prsnt papr, w think of introducing othr diagrams by mploying th composition of th application in [8] with th application in [9] on th main diagram (A) Th following qustions wr posd: 1 Can w find th nw partition from th old on dirctly? Is th movmnt of th bads in th nw main diagrams rgular or not? If it is rgular, can w dsign th nw main diagrams for th guids b, b 3,, and b dpnding on th nw main diagram for b 1? 3 Is thr any rlation btwn th intrsction of th diagrams in th normal cas and th nw cas? To answr ths qustions, th papr is organizd as follows In sction two, w suggst th background and notations In sction thr, w put forth th nw diagrams of (upsid-down o dirct rotation β-numbrs In sction four,
Amrican Journal of Mathmatics and Statistics 014, 4(): 58-64 59 w summariz th ruls for dsigning th nw main diagrams for th guids b,b 3,, and b dpnding on th nw main diagram for b 1 Background and Notations 1 Diagram (A) of β-numbrs Lt r b a non-ngativ intgr, A partition μ = ( μ 1, μ, μ n ) of r is a squnc of non - ngativ intgrs n such that μ = i=1 μ i = r and μ i μ i+1 ; i 1; [3] For xampl, μ = (5, 4, 4,,,,1) is a partition of r =0 β-numbrs was dfind by; s Jams in []: "Fix μ is a partition of r, choos an intgr b gratr than or qual to th numbr of parts of μ and dfin β i = μ i + b i, 1 i b Th st { β 1, β,, β b } is said to b th st of β -numbrs for μ" For th abov xampl, if w tak b =7, thn th st of β-numbrs is{11, 9, 8,5,4, 3,1} Now, lt b a positiv intgr numbr gratr than or qual to, w can rprsnt β - numbrs by a diagram calld diagram (A) runnr1 0 runnr 1 +1 +1 runnr -1-1 3-1 Whr vry β will b rprsntd by a bad ( ) which taks its location in diagram (A) Rturning to th abov xampl, diagram (A) of β -numbrs for = and =3 is as shown blow in diagram 1 and rspctivly: = b = 7 0 1 3 4 5 6 7 8 9 10 11 Diagram 1 Diagram diagram(a) = 3 b = 7 0 1 3 4 5 6 7 8 9 10 11 Not: Throughout this papr, dnots a fixd intgr gratr than or qual to w man by diagram (A); diagram (A) of β-numbrs Th Main Diagrams (A) Mahmood in [6] introducd th dfinition of main diagram(s) (A) and th ida of th intrsction of ths main diagrams in th following subsctions, w rpat th principals rsults, as follows: Sinc th valu of b n; [5], thn w dal with an infinit numbrs of valus of b Hr w want to mntion that for ach valu of b thr is a spcial diagram (A) of β - numbrs for it, but thr is a rpatd part of on's diagram with th othr valus of b whr a "Down shiftd" or "Up- shiftd", occurs whn w tak th following: (b 1 if b = n), (b if b = n+1), and (b if b = n+(-1)) Dfinition (1): [6] Th valus of b 1, b, and b ar calld th guids of any diagram (A) of β -numbrs From th abov xampl whr μ = (5, 4, 4,,,,1), r = 0, if = thn thr ar two guids, th first is b 1 = 7 sinc n = 7 and th scond is b = 8, th β - numbrs ar givn in tabl 1: Tabl 1 β- Numbrs β i b s β 1 β β 3 β 4 β 5 β 6 β 7 β 8 b 1 = 7 11 9 8 5 4 3 1 b = 8 1 10 9 6 5 4 0 W dfin any diagram (A) that corrsponds any b guids as a "main diagram" or "guid diagram" Thorm (): [6] Thr is of main diagrams for any partition μ of r Hnc, for our xampl, w hav two main diagrams for = as shown in diagram 3: = b 1 = 7 b 1 = 8 0 1 3 4 5 6 7 8 9 10 11 1 13 Diagram 3 And th ida of "Down shiftd" or "Up- shiftd", is dclard in diagram 4 blow: = b 1 =7 b 1 +1() b 1 +() 0 1 3 4 5 6 7 8 9 10 11 1 13 14 15 = b = 8 b +1() b +() 0 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 17 Diagram 4 Illustrats th Ida of "Down- shiftd"
60 Ammar Sddiq Mahmood t al: (Upsid-Down o Dirct Rotation) β - Numbrs 3 Som Kinds of Partition Any partition μ of r is calld w-rgular; w, if thr dos not xist i 1 such that μ i = μ i+w 1 > 0, and μ is calld w-rstrictd if μ i μ i+1 < ww; i 1, [3] From th abov xampl, whr μ = ( 5,4,4,,,,1) thn μ is 4-rgular and 3-rstrictd 4 Th Intrsction of th Main Diagrams Th ida of th intrsction of any main diagrams is dfind by th following: 1 Lt τ b th numbr of rdundant part of th partition μ τ of r, thn w hav: μ = (μ 1, μ,, μ n ) = (λ 1 τ 1, λ, τ, λ m n m ) such that μ i = m τ λ j i=1 j=1 j W dnot th intrsction of main diagrams by s=1 m d bs 3 Th intrsction rsult as a numrical valu will b dnotd by # s=1 m d bs, and it is qual to ϕ in th cas of no xistnc of any bad, or γ in th cas that γ common bads xist in th main diagrams For our xampl, th intrsction of th two main diagrams is as shown in diagram 5: b 1 = 7 b = 8 ss=11 mm dd bbss # m d bs s=1 valu if < h or ( = h and h < w), = φ if > h or ( = h and h w) Also, S M Mahmood in [7] gav th sam subjct by using a nw tchniqu which supportd th rsults of Mahmood in [6] Also, S M Mahmood in [7] gav th sam subjct by using a nw tchniqu which supportd th rsults of Mahmood in [6] 3 (Upsid-Down o Dirct Rotation) β - Numbrs In th prsnt papr, w introduc som nw diagrams dpnding on th old diagram (A) by mploying th composition of upsid-down application with dirct rotation application of thr diffrnt dgrs namly 90 o, 180 o and 70 o rspctivly As a prliminary stp toward th subjct, w giv th following notations: 1 By dirct rotation; w man: countr clockwis rotation All th rotations ar about th origin 3 Composition is th combination of two or mor mappings to form a singl nw mapping Hr, w rmind with th dfinition of composition of two mappings: Lt f : S T and g: T U b two mappings W dfin th composition of f followd by g, dnotd by g ο f, to b th mapping (g ο f)(x) = g (f (x)), for all x S Not carfully that in th notation (g ο f ) th mapping on th right is applid first S figur 1 Diagram 5 Notic that, # s=1 m d bs =3 Th two principl thorms about th ida of th intrsction of any main diagrams ar: Thorm (41): [6] For any, th following holds: 1- # s=1 m d bs = ϕ if τ k = 1, k whr 1 k m - Lt Ω b th numbr of parts of λ which satisfis th condition τ k for som k, thn: Ω t=1 # s=1 m d bs = [ τ t Ω ( 1)] Thorm (4): [6] 1- Lt μ b a partition of r and μ is w-rgular, thn: # m d bs s=1 valu if < w, = ϕ if w - Lt μ b a partition of r and μ is h-rstrictd, thn: Figur 1 4 Th nw diagrams cratd by th composition application hav anothr partitions of th origin partition and if w us th ida of th intrsction, th partition of th bads will not b th sam (or will not b th sum) in # s=1 m d bs in th normal main diagrams To raliz ths facts, w study th composition application for ach dgr apart on th prvious xampl whr µ = (5, 4, 3, 1) for = and =3 as follows: 31 (Upsid-Down o Dirct Rotation of dgr 90 o ) β - Numbrs Th diagrams introducd by this application is dnotd by (A 1 ) and ar shown in diagram 6
Amrican Journal of Mathmatics and Statistics 014, 4(): 58-64 61 b 1 = 7 b = 8 Diagram 3 (A) (U-D o R 90 ) on (-) in right 1 st row in th cas b and to add on () in lft last row in th cas b 1 1 st row in th cas b and to add on () in lft (-1) row in th cas b and to add on () in lft This rul is clarifid in diagram 9 For th abov xampl, whr μ = (5, 4, 3, 1) and = 3 b 1 =7 b =8 Diagram 6 (A 1 ) Now, if w us th old tchniqu for finding any partition of any diagram (A 1 ), th valu of th partition will not b qual to th origin partition? so, w dlt any ffct of (-) in (A) aftr th position of β 1, and w start with numbr 1 for th first (-) a (lft to right) in any row xist in (A), and with numbr for th scond (-) and,tc, and w stop with last (-) bfor th position β 1 in (A) as shown in diagram 7 Now, to apply "upsid-down o dirct rotation of dgr 90 o " on (A), th nw vrsion (A 1 ) has th sam partition of (A), s diagram 8 b 1 = 7 b = 8 1 3 4 5 1 3 4 5 x (U-D o R 90 ) Diagram 7 (A) b 1 = 7 b = 8 1 3 5 4 Diagram 8 (A 1 ) 4 1 3 5 x Rmark (311): Th main diagram (A 1 ) in cas b 1 = n, plays a main rol to dsign all th main diagrams (A 1 ) for (b = n+1), and (b = n+(-1)), as follows: Rul (31): Sinc th main diagram (A 1 ) in th cas b 1, w can find th succssiv main diagrams (A 1 ) for b, b 3, and b, as follows: 1 1 st row in th cas b 1 = n nd row in th cas b and to on (-) in right 3 rd row in th cas b 3 and to add on (-) in right last row in th cas b and to add on (-) in right of main diagram (A 1 ) nd row in th cas b 1 3 rd row in th cas b and to add on (-) in right last row in th cas b -1 and to add Diagram b 1 = 7 b = 8 b 3 = 9 Diagram 9 Thorm (313): All th rsults in [6] about th main diagram (A) is th sam of th diagram (A 1 ) but in (upsid-down o dirct rotation of dgr 90 o ) position On of ths rsults is th intrsction of th main diagrams so, th fact mntiond in thorm (313) is clar in diagram 10 comparing it with diagram 5, for = and for =3, s th two diagrams 11 and 1: b 1 = 7 b = 8 ss=11 mm dd bbss Diagram 10 Th intrsction of th main diagrams (A 1 ) for = Notic that, #( s=1 m d bs ) = 3, in both cass b 1 = 7 b = 8 b 3 = 9 ss=11 mm dd bbss Diagram 11 Th intrsction of th main diagrams (A) for =3 b 1 = 7 b = 8 b 3 = 9 ss=11 mm dd bbss b 1 = 7 Diagram 1 Th intrsction of th main diagrams (A 1 ) for =3 3 Notic that, #( s=1 m d bs ) = 1, in both cass
6 Ammar Sddiq Mahmood t al: (Upsid-Down o Dirct Rotation) β - Numbrs 3 (Upsid-Down o Dirct Rotation of dgr 180 o ) β - Numbrs Th diagrams introducd by this application is dnotd by (A ) and ar shown in diagram 13 b 1 =7 b = 8 Diagram 3 (A) Diagram 13 (A ) Again, if w us th old tchniqu for finding any partition of any diagram (A ), th valu of th partition will not b qual to th origin partition? so, w dlt any ffct of (-) in (A) aftr th position of β 1, and w start with numbr 1 for th first (-) a (lft to right) in any row xist in (A), and with numbr for th scond (-) and, tc, and w stop with last (-) bfor th position β 1 in (A) as shown in diagram 7 Now, to apply "upsid-down o Dirct rotation of dgr 180 5 " on (A), th nw vrsion (A ) has th sam partition of (A), s diagram 14 b 1 =7 b = 8 1 3 4 5 1 3 4 5 x b 1 =7 b = 8 Diagram 7 (A) Diagram 14 (A ) b 1 =7 b = 8 1 4 3 5 1 3 4 5 x Rmark(31): Th main diagram (A ) in cas b 1 = n, plays a main rol to dsign all th main diagrams (A ) for (b = n+1), and (b = n+(-1)), as follows: Rul (3): Sinc th main diagram (A ) in th cas b 1, w can find th succssiv main diagrams (A ) for b, b 3, and b, as follows: 1 1 st column in th cas b 1 = n last column in th cas b and to add on () in up (-1) column in th cas b 3 and to add on () in up nd column in th cas b and to add on () in up of main diagram (A ) nd column in th cas b 1 1 st column in th cas b and to add on (-) in down last column in th cas b 3 and to add on () in up 3 rd column in th cas b and to add on () in up last column in th cas b 1 (-1) column in th cas b and to add on (-) in down 1 st column in th cas b and to add on (-) in down To chck this rul For our xampl, whr µ = (5, 4, 3, 1) and = 3, s diagram 15 blow: b 1 = 7 Diagram b 1 = 7 b = 8 b 3 = 9 Diagram 15 Thorm (): All th rsults in [6] about th main diagram (A) is th sam of th diagram (A ) but in (upsid-down o dirct rotation of dgr 180 o ) position Now, as w said bfor, th intrsction of th main diagrams is on of ths rsults, hnc s diagram 16 and compar it with diagram 5 for = and for =3, s diagram 17and compar it with diagram 11 abov: b 1 = 7 b = 8 ss=11 mm dd bbss Diagram 16 Th intrsction of th main diagrams (A ) for = Again, #( s=1 m d bs ) = 3, in both cass b 1 = 7 b = 8 b 3 = 9 ss=11 mm dd bbss Diagram 17 Th intrsction of th main diagrams (A ) for =3 3 Also, #( s=1 m d bs ) = 1, in both cass (Upsid-Down o Dirct Rotation of Dgr 70 o ) β - Numbrs Th diagrams introducd by this application is dnotd by (A 3 ) and ar shown in diagram 18
Amrican Journal of Mathmatics and Statistics 014, 4(): 58-64 63 b 1 = 7 b = 8 Diagram 3 (A) (U-D o R 70 ) b 1 =7 b =8 Diagram 18 (A 3 ) Again, if w us th old tchniqu for finding any partition of any diagram (A 3 ), th valu of th partition will not b qual to th origin partition? so, w dlt any ffct of (-) in (A) aftr th position of β 1, and w start with numbr 1 for th first (-) a (lft to right) in any row xist in (A), and with numbr for th scond (-) and,tc, and w stop with last (-) bfor th position β 1 in (A) as shown in diagram 7 Now, to apply "upsid-down o Dirct rotation of dgr 70 5 " on (A), th nw vrsion (A 3 ) has th sam partition of (A), s diagram 19 b 1 = 7 b = 8 1 3 4 5 1 3 4 5 x Diagram 7 (A) (U-D o R 70 ) b 1 =7 b =8 4 X 5 3 1 5 3 1 4 Diagram 19 (A 3 ) Rmark (1): Th main diagram (A 3 ) in cas b 1 = n, plays a main rol to dsign all th main diagrams (A 3 ) for (b = n+1), and (b = n+(-1)), as follows: Rul (): Sinc th main diagram (A 3 ) in th cas b 1, w can find th succssiv main diagrams (A 3 ) for b, b 3, and b, as follows: 1 1 st row in th cas b 1 = n last row in th cas b and to add on () in right (-1) row in th cas b 3 and to add on () in right nd row in th cas b and to add on () in right of main diagram (A 3 ) nd row in th cas b 1 1 st row in th cas b and to add on (-) in lft last row in th cas b 3 and to add on () in right 3 rd row in th cas b and to add on () in right ) last row in th cas b 1 (-1) row in th cas b and to add on (-) in lft 1 st row in th cas b and to add on (-) in lft To matrializ rul () For th our xampl for = 3, s diagram 0: b 1 = 7 Diagram b 1 = 7 b = 8 b 3 = 9 Diagram 0 Thorm (3): All th rsults in [6] about th main diagram (A) is th sam of th diagram (A 3 ) but in (upsid-down o dirct rotation of dgr 70 o ) position To prciv thorm (3) for this typ of rotation, on our xampl, obsrv diagrams 1 and compar it with diagram 5 for = and diagram to b compard with diagram 11 for =3: b 1 = 7 b = 8 ss=11 mm dd bbss Diagram 1 Th intrsction of th main diagrams (A 3 ) for = Notic that, #( s=1 m d bs ) = 3, in both cass b 1 = 7 b = 8 b 3 = 9 ss=11 mm dd bbss Diagram Th intrsction of th main diagrams (A 3 ) for =3 3 Also, #( s=1 m d bs ) = 1, in both cass 4 Conclusions 1 A procdur is suggstd for th diagrams (A 1 ), (A ) and (A 3 ) of β - numbrs which thy rprsnt th composition of upsid - down application with th dirct rotation application of dgrs 90 o, 180 o, and 70 o rspctivly, on diagram (A) of β-numbrs to hav th sam partition of diagram (A) of β-numbrs Furthrmor, for ach composition, a rul for dsigning
64 Ammar Sddiq Mahmood t al: (Upsid-Down o Dirct Rotation) β - Numbrs all th main diagrams of th composition for b, b 3,, and b is st dpnding on th main diagram of th composition for b 1 3 W find out that th intrsction of th main diagrams of ach composition is th sam of th main diagram (A) but in th composition position 4 And finally: a) (Upsid-Down o Dirct Rotation of dgr 90 o ) β - Numbrs = (Dirct Rotation of dgr 70 o o Upsid-Down) β - Numbrs b) (Upsid-Down o Dirct Rotation of dgr 180 o ) β - Numbrs = (Dirct Rotation of dgr 180 o o Upsid-Down) β - Numbrs c) (Upsid-Down o Dirct Rotation of dgr 70 o ) β - Numbrs = (Dirct Rotation of dgr 90 o o Upsid-Down) β - Numbrs REFERENCES [1] Gorg E Andrws, Th Thory of Partitions, Encyclopdia of Mathmatics and its Applications, Vol, Addison-Wsly publishing company, London, 1976 [] G Jams, Som combinatorial rsults involving Young diagrams, Math Proc Cambridg Philos Soc,83,1-10, 1978 [3] A Mathas, Iwahori-Hck Algbras and Schur Algbras of th Symmtric Groups, Amr Math Soc Univrsity Lctur Sris, 15, 1999 [4] M Fayrs, Anothr runnr rmoval thorm for r- dcomposition numbr of lwahori Hck algbras and q- Schur algbras, J algbra, 310, 396-404,007 [5] H S Mohammad, Algorithms of Th Cor of Algbraic Young's Tablaux, M Sc Thsis, Collg of Education, Univrsity of Mosul, 008 [6] A S Mahmood, On th intrsction of Young's diagrams cor, J Education and Scinc (Mosul Univrsity), 4, no 3, 143-159, 011 [7] S M Mahmood, On - Rgular and th Intrsction of Young's Diagrams Cor, M Sc Thsis, Collg of Education, Univrsity of Mosul, 011 [8] A S Mahmood, Upsid - down β - numbrs, Australian J of Basic and Applid Scincs 7, no 7, 986-990, 013 [9] A S Mahmood, and Sh S Ali, Dirct Rotation β- numbrs, Journal of Advancs in Mathmatics, Vol 5, No, 64-650, 013