Guerino Mazzola U & ETH Zürich Internet Institute for Music Science Classification Theory and Universal Constructions in Categories of Musical Objects Classification Theory and Universal Constructions in Categories of Musical Objects

Contents Enumeration of classical objects Local classification techniques Globalization and general addresses Resolutions for global classification Enumeration of classical objects Local classification techniques Globalization and general addresses Resolutions for global classification

Enumeration C  Ÿ  (chords) M  – 2 (motives) Enumeration = calculation of the number of orbits of a set C of such objects under the canonical left action G ¥ C  C of a subgroup G  Aff*(F) =  e F  GL(F) Ambient module F = Ÿ   – 2 in the above examples

Enumeration 1973 A. Forte (1980 J.Rahn) List of 352 orbits of chords under the translation group T 12 = e Ÿ  and the group TI 12 = e Ÿ . ± 1 of translations and inversions on Ÿ  1978 G. Halsey/E. Hewitt Recursive formula for enumeration of translation orbits of chords in finite abelian groups F Recursive formula for enumeration of translation orbits of chords in finite abelian groups F Enumeration of orbit numbers for chords in cyclic groups Ÿ n, n c 24 Enumeration of orbit numbers for chords in cyclic groups Ÿ n, n c G. Mazzola List of the 158 affine orbits of chords in Ÿ  List of the 158 affine orbits of chords in Ÿ  List of the 26 affine orbits of 3-elt. motives in ( Ÿ   2 and 45 in Ÿ  ¥  Ÿ  List of the 26 affine orbits of 3-elt. motives in ( Ÿ   2 and 45 in Ÿ  ¥  Ÿ  1989 H. Straub /E.Köhler List of the 216 affine orbits of 4-element motives in ( Ÿ   H. Fripertinger Enumeration formulas for T n, TI n, and affine chord orbits in Ÿ n, n-phonic k-series, all-interval series, and motives in Ÿ n  ¥  Ÿ m Enumeration formulas for T n, TI n, and affine chord orbits in Ÿ n, n-phonic k-series, all-interval series, and motives in Ÿ n  ¥  Ÿ m Lists of affine motive orbits in ( Ÿ   2 up to 6 elements, explicit formula... Lists of affine motive orbits in ( Ÿ   2 up to 6 elements, explicit formula...

Enumeration x^144 + x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^ x^72 + …. … x x x x 2 + x + 1 = cycle index polynomial x 72 ª x 72 average # of stars in a galaxis =

Enumeration Polya-de-Bruijn theory: Cycle index polynomial Identify subsets C  F (usually F = Ÿ n ) with their characteristic function  C : F  For a permutation g in the group G  Aff*(F), we have cycle index cyc(g) = (c1,…cf), f = #F Take indeterminates X 1,…X f, set X g = X 1 c1... X f cf  G Cycle index polynomial is Z(G) = (#G) -1  G X g

Enumeration Polya-de-Bruijn theory: Configuration counting series Consider polynomial Polya weights w(0), w(1) in – [x] For  : F , we have G-invariant product  w  =  F w(  (t))  2 F /G The configuration counting series is C(G,F,w) =  2 F /G  w 

Enumeration Facts For w(0) = 1, w(1) = x, the x k coefficient of C(G,F,w) is the number of G-orbits of k-element sets in F For w(0) = 1, w(1) = x, the x k coefficient of C(G,F,w) is the number of G-orbits of k-element sets in F For the constant weight w(0) = w(1) = 1, C(G,F,w) = # 2 F /G (Main) Theorem C(G,F,w) = Z(G)(w(0)+w(1), w(0) 2 +w(1) 2,…,w(0) f +w(1) f ) Corollary C(G,F,w) = Z(G)(w(0)+w(1), w(0) 2 +w(1) 2,…,w(0) f +w(1) f ) Corollary # 2 F /G = Z(G)(2,2,…,2) # 2 F /G = Z(G)(2,2,…,2)

Enumeration From generalizations of the main theorem by N.G. de Bruijn, we have (for example) the following enumerations: k = T TI Aff*( Ÿ 12 ) k# of orbits of (k,12)-series k# of orbits of (k,12)-series (dodecaphonic)

Local Techniques Categories of local compositions Fix commutative ring R For any two (left) R-modules A,B, let = e B.Lin(A,B) be the R-module of R-affine morphisms F(a) = e b.F 0 (a) = b + F 0 (a) F 0 = linear part, e b = translation part The category Loc R of local compositions over R has as objects the couples (K,A) of subsets K of R-modules A, and as morphisms f: (K,A)  L,B) set maps f: K  L which are induced by affine morphism F in

Local Techniques Local Classification: Calculate the isomorphism classes in Loc R ! For finite local compositions, we have this procedure: Represent (K,A) by an affine k: R n  A with K = {k(e 0 ),k(e 1 ),…,k(e n )} for #K = n+1 Then identify K to the orbit k. S n+1 of the right action of the symmetric group of the affine basis e 0 = 0, e 1,..., e n Get rid of the translations within A by taking the linear part k 0 of k, corresponding to the passage to the difference dK = {k(e 1 ) - k(e 0 ),…,k(e n ) - k(e 0 )} Take the induced right linear action of S n+1 and the left action of GL(A) on Lin(R n,A).

Local Techniques Proposition Let Gen(R n,A)  Lin(R n,A) be the subset of difference maps dk: R n  A with dk = surjective dk = surjective dk(e s ) π 0 and dk(e s ) π dk(e t ) for all s π t. dk(e s ) π 0 and dk(e s ) π dk(e t ) for all s π t. Take the induced right linear action of S n+1 and the left action of GL(A) on Gen(R n,A) Take the induced right linear action of S n+1 and the left action of GL(A) on Gen(R n,A) Let LoClass(A,n+1,R) be the set of isomorphism classes of local compositions K of cardinality n+1 and ambient space A local compositions K of cardinality n+1 and ambient space A K is generating, i.e. A = R.K =  s,t  R.(k s - k t ) K is generating, i.e. A = R.K =  s,t  R.(k s - k t ) Then we have a canonical bijection GL(A)\Gen(R n,A)/ S n+1

Local Techniques Let X n (R,A) be the set of submodules V  R n with R n A R n A e s, e s -e t œ V for all s π t together with the above right action of S n+1 e s, e s -e t œ V for all s π t together with the above right action of S n+1 Sending dk: R n  A to ker(dk) induces a bijection GL(A)\Gen(R n,A)/ S X n (R,A)/ S n+1 GL(A)\Gen(R n,A)/ S X n (R,A)/ S n+1 Let X n (R,r) be the S n+1 -stable set of submodules V  R n with R n /V = locally free of rank r R n /V = locally free of rank r e s, e s -e t œ V for all s π t e s, e s -e t œ V for all s π t X n (R,r) =  Grass n,r (R) - V n (R,r) V n (R,r) = closed, S n+1 -stable subscheme of Grass n,r (R)

Local Techniques Theorem (local geometric classification) There is a quotient scheme, i.e., an exact sequence X n,r ¥S n+1 X n,r ¥ S n+1 X n,r X n,r /S n+1 pr 1  Its R-valued points are the orbits of X n,r, and if R is semi-simple, X n,r (R) = X n (R,r)

Local Techniques Application to orbit algorithms for rings Application to orbit algorithms for rings R of finite length R of finite length R local R local self-injective self-injective E.g. R = E.g. R = Ÿ s n, s = prime subspace V  R n subgroup G  S n+1 subspace V  R n subgroup G  S n+1 soc(R n )  V V/soc (R n )  R/soc(R)) n soc(V) π  soc(R n ) V = soc(V) V   R/Rad(R)) n soc(V) π  soc(R n ) V π soc(V) I(V)  R n (direct factor) I(V) R m m < n R m m < n G := Iso(I(V)) V  R m I(V)  R n (direct factor) I(V) R m m < n R m m < n G := Iso(I(V)) V  R m

Local Techniques Classes of 3-element motives M  ( Ÿ 12 ) generic

K Globalization local iso  C i  K i  K t  C t K it  K ti 

Globalization I IV II VI V III VII The class nerve cn(K) of global composition is not classifying

Globalization M  B M A = address of the composition M  M  (A= R)

Classify! Globalization The category Loc A of local A-addressed compositions has as objects the couples (K, of sets K of affine morphisms in and as morphisms f: (K,  L, set maps f: K  L which are naturally induced by affine morphism F in The category Glob A of global A-addressed compositions has as objects K I coverings of sets K by atlases I of local A- addressed compositions with manifold gluing conditions and manifold morphisms f  : K I  L J, including and compatible with atlas morphisms  : I  J

Resolutions Standard A-addressed local compositions 0 c n, A + n = R n ≈ A n+1 e i canonical linear basis of R n a  A,  0  c i c n, a i = (0,…,a,…,0)  i : A  A + n  0 (a) = (0, a 0 ) i = 0  i (a) = (e i, a i ) 0 < i A  n  + n AnAnAnAn s. S s 0, s 1,…, s n  S = {s 0, s 1,…, s n }  s 0, s 1,…, s n  S = {s 0, s 1,…, s n }  s. : A  n  S:  i  s i s 0, s 1,…, s n  S = {s 0, s 1,…, s n }  s 0, s 1,…, s n  S = {s 0, s 1,…, s n }  s. : A  n  S:  i  s i universal property

Resolutions Standard A-addressed global compositions n* = covering of an interval [0,m] by non-empty subsets A  n* n* induces a A-addressed global standard composition A  n* A  m which is canonically deduced from A  m by the n*-charts Choose enumeration K = {k 0,k 1,…,k m } of the global compostition K I. Call n*(K I ) the covering of [0,m] corresponding to the atlas I.  K I = A  n*(K I ) is the resolution ofK I  K I = A  n*(K I ) is the resolution of K I res(K I ) :  K I  K I A  m i  K i A  m i  K i res(K I ) :  K I  K I A  m i  K i A  m i  K i universal morphism, natural in K I  

Resolutions Represent K I by module complexes of function in  K I A module complex M on K I is a „coefficient system“ on the nerve n(K I ), i.e. a functor M(  ) = R-module for simplex  affine transition morphisms M  : M(  )  M(  ),     Important Examples: Function complex n  (K I ): For simplex , take local composition   K i n  (K I )(  ) = Loc R ( , N  f   n  ( L J )  n  (K I ) subcomplex; for morphism f  : L J  K I  of global compositions, we have induced function complex N  f   n  ( L J )

Resolutions The resolution complex of K I res(K I ):  K I  K I  n  (K I ) = n  (K I )  res(K I ) Classification Strategy: Reconstruct K I from its resolution complex Reconstruct K I from its resolution complex Classify a relevant set of module complexes Classify a relevant set of module complexes Res A,n* = {N  n  ( A  n* ), properties…} which relate to resolution complexes of global compositions K I with n*(K I n*

Resolutions Reconstruction of K I e N,  (s) (a)(l) = l(s)(a) Local construction for simplex  of the resolution A  n* and module N(  ): e N,  : A     )* e N,  (s) (a)(l) = l(s)(a) for a function l in N(  ) Have local compositions  /N  =  Im(e N,  )   )*, and canonical local morphisms  /N  /N for   A  n* /N  =  colim n*  /N Global construction: A  n* /N  =  colim n*  /N

Resolutions Proposition 1. Let K I have these properties (*) the chart modules R.K i are projective of finite type the chart modules R.K i are projective of finite type the function modules n  (K I )  K i ) are projective the function modules n  (K I )  K i ) are projective Then A  n* /  n  (K I )  K I 2. Res A,n* = {N  n  ( A  n* ), Const  N  0 ) = projective of finite type N separates points of A  n* } Then we have a canonical bijection Res A,n* /Aut( A  n* IsoClasses[K I with (*) and n*(K I n*]

Resolutions Theorem (global addressed geometric classification) Let A be locally free of a defined rank. Then there is a subscheme J n* of a projective R-scheme of finite type such that its S-valued points a subscheme J n* of a projective R-scheme of finite type such that its S-valued points  : Spec(S)  J n* are in bijection with the classifying orbits of module complexes N in S ƒ R A  n* which are locally free of defined co-ranks on the zero simplexes of n*. Res n*,r ¥ Aut( A  n* ) Res n*,r Res n*,r / Aut( A  n* ) = J n* pr 1 