Vector-partition functions Matthias Beck San Francisco State - PowerPoint PPT Presentation

Vector-partition functions Matthias Beck San Francisco State University math.sfsu.edu/beck

Vector partition functions A – an ( m × d ) -integral matrix b ∈ Z m � x ∈ Z d � Goal: Compute vector partition function φ A ( b ) := # ≥ 0 : A x = b (defined for b in the nonnegative linear span of the columns of A ) Vector-partition functions Matthias Beck 2

Vector partition functions A – an ( m × d ) -integral matrix b ∈ Z m � x ∈ Z d � Goal: Compute vector partition function φ A ( b ) := # ≥ 0 : A x = b (defined for b in the nonnegative linear span of the columns of A ) Applications in... ◮ Number Theory (partitions) ◮ Discrete Geometry (polyhedra) ◮ Commutative Algebra (Hilbert series) ◮ Algebraic Geometry (toric varieties) ◮ Representation Theory (tensor product multiplicities) ◮ Optimization (integer programming) ◮ Chemistry, Biology, Physics, Computer Science, Economics... Vector-partition functions Matthias Beck 2

An example � � 1 2 1 0 A = 1 1 0 1 Vector-partition functions Matthias Beck 3

An example � � 1 2 1 0 A = 1 1 0 1 � � a �� x ∈ Z 4 φ A ( a, b ) = # ≥ 0 : A x = b  4 + a + 7+( − 1) a a 2 if a ≤ b  8   2 + 7+( − 1) a ab − a 2 4 − b 2 2 + a + b if a = 2 − 1 ≤ b ≤ a + 1 8 b 2  2 + 3 b if b ≤ a 2 + 1   2 Vector-partition functions Matthias Beck 3

An example a � 2 − 1 ≤ b ≤ a + 1 b ✻ � � � � 1 2 1 0 � a ≤ b � � A = ✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟ � ✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟ � 1 1 0 1 � � � � � � � � � � � b ≤ a � � � 2 � � � � ✲ � � a � � � � � � � � a �� x ∈ Z 4 φ A ( a, b ) = # ≥ 0 : A x = b  4 + a + 7+( − 1) a a 2 if a ≤ b  8   2 + 7+( − 1) a ab − a 2 4 − b 2 2 + a + b if a = 2 − 1 ≤ b ≤ a + 1 8 b 2  2 + 3 b if b ≤ a 2 + 1   2 Vector-partition functions Matthias Beck 3

A “one-dimensional” example A = ( a 1 , a 2 , . . . , a d ) Vector-partition functions Matthias Beck 4

A “one-dimensional” example A = ( a 1 , a 2 , . . . , a d ) Restricted partition function ( m 1 , . . . , m d ) ∈ Z d � � φ A ( t ) = # ≥ 0 : m 1 a 1 + · · · + m d a d = t , a quasi-polynomial, i.e., φ A ( t ) = c d − 1 ( t ) t d − 1 + c d − 2 ( t ) t d − 2 + · · · + c 0 ( t ) where c k ( t ) are periodic Vector-partition functions Matthias Beck 4

A “one-dimensional” example A = ( a 1 , a 2 , . . . , a d ) Restricted partition function ( m 1 , . . . , m d ) ∈ Z d � � φ A ( t ) = # ≥ 0 : m 1 a 1 + · · · + m d a d = t , a quasi-polynomial, i.e., φ A ( t ) = c d − 1 ( t ) t d − 1 + c d − 2 ( t ) t d − 2 + · · · + c 0 ( t ) where c k ( t ) are periodic Frobenius problem: find the largest value for t such that φ A ( t ) = 0 Vector-partition functions Matthias Beck 4

Ehrhart quasi-polynomials Rational (convex) polytope P – convex hull of finitely many points in Q d x ∈ R d : A x ≤ b � � Alternative description: P = � t P ∩ Z d � For t ∈ Z > 0 , let L P ( t ) := # Vector-partition functions Matthias Beck 5

Ehrhart quasi-polynomials Rational (convex) polytope P – convex hull of finitely many points in Q d x ∈ R d : A x ≤ b � � Alternative description: P = � � x ∈ R d Translate & introduce slack variables − → P = ≥ 0 : A x = b � t P ∩ Z d � For t ∈ Z > 0 , let L P ( t ) := # Vector-partition functions Matthias Beck 5

Ehrhart quasi-polynomials Rational (convex) polytope P – convex hull of finitely many points in Q d x ∈ R d : A x ≤ b � � Alternative description: P = � � x ∈ R d Translate & introduce slack variables − → P = ≥ 0 : A x = b � t P ∩ Z d � For t ∈ Z > 0 , let L P ( t ) := # = φ A ( t b ) (for fixed b ) Vector-partition functions Matthias Beck 5

Ehrhart quasi-polynomials Rational (convex) polytope P – convex hull of finitely many points in Q d x ∈ R d : A x ≤ b � � Alternative description: P = � � x ∈ R d Translate & introduce slack variables − → P = ≥ 0 : A x = b � t P ∩ Z d � For t ∈ Z > 0 , let L P ( t ) := # = φ A ( t b ) (for fixed b ) Theorem (Ehrhart 1967) If P is a rational polytope, then the functions L P ( t ) and L P ◦ ( t ) are quasi-polynomials in t of degree dim P . If P has integer vertices, then L P and L P ◦ are polynomials. Furthermore, L P (0) = 1 Theorem (Ehrhart, Macdonald 1970) L P ( − t ) = ( − 1) dim P L P ◦ ( t ) Vector-partition functions Matthias Beck 5

Corollaries due to Ehrhart theory The computation of the (Ehrhart-)quasi-polynomial ( m 1 , . . . , m d ) ∈ Z d � � φ A ( t ) = # ≥ 0 : m 1 a 1 + · · · + m d a d = t , gives rise to the Fourier-Dedekind sum (M B–Robins 2003) λ n σ n ( a 1 , . . . , a d ; a 0 ) := 1 � (1 − λ a 1 ) · · · (1 − λ a d ) . a 0 λ a 0 =1 Vector-partition functions Matthias Beck 6

Corollaries due to Ehrhart theory The computation of the (Ehrhart-)quasi-polynomial ( m 1 , . . . , m d ) ∈ Z d � � φ A ( t ) = # ≥ 0 : m 1 a 1 + · · · + m d a d = t , gives rise to the Fourier-Dedekind sum (M B–Robins 2003) λ n σ n ( a 1 , . . . , a d ; a 0 ) := 1 � (1 − λ a 1 ) · · · (1 − λ a d ) . a 0 λ a 0 =1 Choosing d = 2 , n = 0 , a 1 = a, a 2 = 1 , a 0 = b gives rise to the classical Dedekind sum b − 1 s ( a, b ) := 1 � πja � � πj � � cot cot 4 b b b j =1 Vector-partition functions Matthias Beck 6

Corollaries due to Ehrhart theory Ehrhart-Macdonald Reciprocity yields the functional identity φ A ( − t ) = ( − 1) d − 1 φ A ( t − ( a 1 + · · · + a d )) Vector-partition functions Matthias Beck 7

Corollaries due to Ehrhart theory Ehrhart-Macdonald Reciprocity yields the functional identity φ A ( − t ) = ( − 1) d − 1 φ A ( t − ( a 1 + · · · + a d )) The identity φ A (0) = 1 implies the Reciprocity Law for Zagier’s “higher- dimensional Dedekind sums” a 0 − 1 � πja 1 � � πja d � s ( a 1 , a 2 , . . . , a d ; a 0 ) := 1 � cot · · · cot . a 0 a 0 a 0 j =1 Vector-partition functions Matthias Beck 7

Corollaries due to Ehrhart theory Ehrhart-Macdonald Reciprocity yields the functional identity φ A ( − t ) = ( − 1) d − 1 φ A ( t − ( a 1 + · · · + a d )) The identity φ A (0) = 1 implies the Reciprocity Law for Zagier’s “higher- dimensional Dedekind sums” a 0 − 1 � πja 1 � � πja d � s ( a 1 , a 2 , . . . , a d ; a 0 ) := 1 � cot · · · cot . a 0 a 0 a 0 j =1 The identity φ A ( t ) = 0 for − ( a 1 + · · · + a d ) < t < 0 gives a new reciprocity relation which is a “higher-dimensional” analog of that for the the Dedekind-Rademacher sum. Vector-partition functions Matthias Beck 7