Flow polytopes in combinatorics and algebra Karola M esz aros - PowerPoint PPT Presentation

More examples of flow polytopes Example x 14 G is the complete graph k n +1 x 24 x 13 a = (1 , 1 , . . . , 1 , − n ) x 12 a x 23 x 34 1 1 1 − 3 Theorem (M, Morales, Rhoades 2014) volume equals # SYT( n − 1 , n − 2 , . . . , 2 , 1) · C 1 C 2 · · · C n − 1 Combinatorial proof?

More examples of flow polytopes Example x 14 G is the complete graph k n +1 x 24 x 13 a = (1 , 1 , . . . , 1 , − n ) x 12 a x 23 x 34 1 1 1 − 3 Theorem (M, Morales, Rhoades 2014) volume equals # SYT( n − 1 , n − 2 , . . . , 2 , 1) · C 1 C 2 · · · C n − 1 Combinatorial proof? Relation to CRY?

Even more examples of flow polytopes Theorem (Postnikov 2013) If G is a planar graph then F G (1 , 0 , . . . , 0 , − 1) is integrally equivalent to an order polytope of a certain poset P G . Corollary If G is a planar graph then vol F G (1 , 0 , . . . , 0 , − 1) = # linear extensions of P G .

Examples of flow polytopes F G ( a ) = { flows x ( ǫ ) ∈ R ≥ 0 , ǫ ∈ E ( G ) | netflow( i ) = a i } Example (Baldoni-Vergne 2008) Π 4 x 1 x 2 x 3 y 1 y 2 y 3 a 1 a 2 a 3 − a 1 − a 2 − a 3

Examples of flow polytopes F G ( a ) = { flows x ( ǫ ) ∈ R ≥ 0 , ǫ ∈ E ( G ) | netflow( i ) = a i } Example (Baldoni-Vergne 2008) Π 4 x 1 x 2 x 3 y 1 y 2 y 3 a 1 a 2 a 3 − a 1 − a 2 − a 3 x 1 + y 1 = a 1 − → x 1 ≤ a 1

Examples of flow polytopes F G ( a ) = { flows x ( ǫ ) ∈ R ≥ 0 , ǫ ∈ E ( G ) | netflow( i ) = a i } Example (Baldoni-Vergne 2008) Π 4 x 1 x 2 x 3 y 1 y 2 y 3 a 1 a 2 a 3 − a 1 − a 2 − a 3 x 1 + y 1 = a 1 − → x 1 ≤ a 1 x 2 + y 2 − y 1 = a 2 − → x 1 + x 2 ≤ a 1 + a 2

Examples of flow polytopes F G ( a ) = { flows x ( ǫ ) ∈ R ≥ 0 , ǫ ∈ E ( G ) | netflow( i ) = a i } Example (Baldoni-Vergne 2008) Π 4 x 1 x 2 x 3 y 1 y 2 y 3 a 1 a 2 a 3 − a 1 − a 2 − a 3 x 1 + y 1 = a 1 − → x 1 ≤ a 1 x 2 + y 2 − y 1 = a 2 − → x 1 + x 2 ≤ a 1 + a 2 x 3 + y 3 − y 2 = a 3 − → x 1 + x 2 + x 3 ≤ a 1 + a 2 + a 3

Examples of flow polytopes F G ( a ) = { flows x ( ǫ ) ∈ R ≥ 0 , ǫ ∈ E ( G ) | netflow( i ) = a i } Example (Baldoni-Vergne 2008) Π 4 x 1 x 2 x 3 y 1 y 2 y 3 a 1 a 2 a 3 − a 1 − a 2 − a 3 x 1 + y 1 = a 1 − → x 1 ≤ a 1 x 2 + y 2 − y 1 = a 2 − → x 1 + x 2 ≤ a 1 + a 2 x 3 + y 3 − y 2 = a 3 − → x 1 + x 2 + x 3 ≤ a 1 + a 2 + a 3 F Π n +1 ( a ) is the Pitman-Stanley polytope

Pitman-Stanley polytope a = ( a 1 , a 2 , . . . , a n ) ∈ Z n ≥ 0 x 1 ≤ a 1 � � x 1 + x 2 ≤ a 1 + a 2 ( x 1 , . . . , x n ) ∈ R n PS n ( a ) = ≥ 0 . . . x 1 + · · · + x n ≤ a 1 + · · · + a n Example PS 2 (1 , 1)

Pitman-Stanley polytope a = ( a 1 , a 2 , . . . , a n ) ∈ Z n ≥ 0 x 1 ≤ a 1 � � x 1 + x 2 ≤ a 1 + a 2 ( x 1 , . . . , x n ) ∈ R n PS n ( a ) = ≥ 0 . . . x 1 + · · · + x n ≤ a 1 + · · · + a n Example PS 2 (1 , 1) PS 3 (1 , 1 , 1)

Pitman-Stanley polytope a = ( a 1 , a 2 , . . . , a n ) ∈ Z n ≥ 0 x 1 ≤ a 1 � � x 1 + x 2 ≤ a 1 + a 2 ( x 1 , . . . , x n ) ∈ R n PS n ( a ) = ≥ 0 . . . x 1 + · · · + x n ≤ a 1 + · · · + a n Example PS 2 (1 , 1) PS 3 (1 , 1 , 1) • 2 n vertices, n dimensional, is a generalized permutahedron

Generalized permutahedra 123 213 132 312 231 321

Volume of the Pitman-Stanley polytope Theorem (Pitman-Stanley 01) � � � n a j 1 1 · · · a j n vol PS n ( a ) = n j 1 , . . . , j n j � (1 ,..., 1) � = a f (1) · · · a f ( n ) f parking function

Volume of the Pitman-Stanley polytope Theorem (Pitman-Stanley 01) � � � n a j 1 1 · · · a j n vol PS n ( a ) = n j 1 , . . . , j n j � (1 ,..., 1) � = a f (1) · · · a f ( n ) f parking function Example vol PS 2 ( a 1 , a 2 ) = 2 a 1 a 2 + a 2 1 = a 1 a 2 + a 2 a 1 + a 2 1

Volume of the Pitman-Stanley polytope Theorem (Pitman-Stanley 01) � � � n a j 1 1 · · · a j n vol PS n ( a ) = n j 1 , . . . , j n j � (1 ,..., 1) � = a f (1) · · · a f ( n ) f parking function Example vol PS 2 ( a 1 , a 2 ) = 2 a 1 a 2 + a 2 1 = a 1 a 2 + a 2 a 1 + a 2 1 2 1 2 1 2 1

Volume of the Pitman-Stanley polytope Theorem (Pitman-Stanley 01) � � � n a j 1 1 · · · a j n vol PS n ( a ) = n j 1 , . . . , j n j � (1 ,..., 1) � = a f (1) · · · a f ( n ) f parking function Example vol PS 2 ( a 1 , a 2 ) = 2 a 1 a 2 + a 2 1 = a 1 a 2 + a 2 a 1 + a 2 1 vol PS 3 ( a 1 , a 2 , a 3 ) = 6 a 1 a 2 a 3 + 3 a 2 1 a 2 + 3 a 1 a 2 2 + 3 a 2 1 a 3 + a 3 1

Volume of the Pitman-Stanley polytope Theorem (Pitman-Stanley 01) � � � n a j 1 1 · · · a j n vol PS n ( a ) = n j 1 , . . . , j n j � (1 ,..., 1) � = a f (1) · · · a f ( n ) f parking function Example vol PS 2 ( a 1 , a 2 ) = 2 a 1 a 2 + a 2 1 = a 1 a 2 + a 2 a 1 + a 2 1 vol PS 3 ( a 1 , a 2 , a 3 ) = 6 a 1 a 2 a 3 + 3 a 2 1 a 2 + 3 a 1 a 2 2 + 3 a 2 1 a 3 + a 3 1 Proof via a subdivision where each term corresponds to the volume of a cell in subdivision

Lattice points of the Pitman-Stanley polytope Theorem (Pitman-Stanley, Gessel 01) � � a 1 t + 1 � � � � a 2 t � � � � a n t � � � L PS n ( a ) ( t ) = · · · j 1 j 2 j n j � (1 ,..., 1)

Lattice points of the Pitman-Stanley polytope Theorem (Pitman-Stanley, Gessel 01) � � a 1 t + 1 � � � � a 2 t � � � � a n t � � � L PS n ( a ) ( t ) = · · · j 1 j 2 j n j � (1 ,..., 1) � � m � � is “ m multichoose k ” n

Lattice points of the Pitman-Stanley polytope Theorem (Pitman-Stanley, Gessel 01) � � a 1 t + 1 � � � � a 2 t � � � � a n t � � � L PS n ( a ) ( t ) = · · · j 1 j 2 j n j � (1 ,..., 1) � � m � � is “ m multichoose k ” n � � 3 � � = 6 , counting { 1 , 1 } , { 1 , 2 } , { 1 , 3 } , { 2 , 2 } , { 2 , 3 } , { 3 , 3 } 2

Lattice points of the Pitman-Stanley polytope Theorem (Pitman-Stanley, Gessel 01) � � a 1 t + 1 � � � � a 2 t � � � � a n t � � � L PS n ( a ) ( t ) = · · · j 1 j 2 j n j � (1 ,..., 1) � � m � � is “ m multichoose k ” n � � 3 � � = 6 , counting { 1 , 1 } , { 1 , 2 } , { 1 , 3 } , { 2 , 2 } , { 2 , 3 } , { 3 , 3 } 2 � � m � � � m + n − 1 � = n n

Lattice points of the Pitman-Stanley polytope Theorem (Pitman-Stanley, Gessel 01) � � a 1 t + 1 � � � � a 2 t � � � � a n t � � � L PS n ( a ) ( t ) = · · · j 1 j 2 j n j � (1 ,..., 1) Corollary L PS n ( a ) ( t ) ∈ N [ t ]

Summary F G ( a ) = { flows x ( ǫ ) ∈ R ≥ 0 , ǫ ∈ E ( G ) | netflow( i ) = a i } Examples • F k n +1 ( a ) : CRY polytope ( a = (1 , 0 , . . . , 0 , − 1) ), Tesler polytope ( a = (1 , 1 , . . . , 1 , − n ) ); volumes divisible by C 1 · · · C n − 2 • F Π n +1 ( a ) : Pitman-Stanley polytope , explicit volume and lattice point formulas related to parking functions. Question • Is there a formula for volume and lattice points of F G ( a ) ?

Lidskii volume formula Theorem (Baldoni-Vergne 08, Postnikov-Stanley - unpublished) G m edges, n + 1 vertices, a i ≥ 0 � m − n � � a j 1 1 · · · a j n vol F G ( a 1 , . . . , a n ) = n j 1 , . . . , j n j � o × K G ( j 1 − o 1 , . . . , j n − o n , 0) where o = ( o 1 , . . . , o n ) , o v = outdeg ( v ) − 1 and | j | = m − n .

Lidskii volume formula Theorem (Baldoni-Vergne 08, Postnikov-Stanley - unpublished) G m edges, n + 1 vertices, a i ≥ 0 � m − n � � a j 1 1 · · · a j n vol F G ( a 1 , . . . , a n ) = n j 1 , . . . , j n j � o × K G ( j 1 − o 1 , . . . , j n − o n , 0) where o = ( o 1 , . . . , o n ) , o v = outdeg ( v ) − 1 and | j | = m − n . Pitman-Stanley polytope: Π 4 � � � n a j 1 1 · · · a j n vol F Π n +1 ( a ) = n · 1 j 1 , . . . , j n j � (1 ,..., 1)

Lidskii volume formula Theorem (Baldoni-Vergne 08, Postnikov-Stanley - unpublished) G m edges, n + 1 vertices, a i ≥ 0 � m − n � � a j 1 1 · · · a j n vol F G ( a 1 , . . . , a n ) = n j 1 , . . . , j n j � o × K G ( j 1 − o 1 , . . . , j n − o n , 0) where o = ( o 1 , . . . , o n ) , o v = outdeg ( v ) − 1 and | j | = m − n . Example � 2 � � 2 � vol F G ( 1 ) = K G (1 − 1 , 1 − 1 , 0) + K G (2 − 1 , 0 − 1 , 0) 1 , 1 2 , 0 = 2 · 1 + 1 · 2 = 4 .

Lidskii volume formula Theorem (Baldoni-Vergne 08, Postnikov-Stanley - unpublished) G m edges, n + 1 vertices, a i ≥ 0 � m − n � � a j 1 1 · · · a j n vol F G ( a 1 , . . . , a n ) = n j 1 , . . . , j n j � o × K G ( j 1 − o 1 , . . . , j n − o n , 0) where o = ( o 1 , . . . , o n ) , o v = outdeg ( v ) − 1 and | j | = m − n . Corollary: vol F G (1 , 0 , . . . , 0 , − 1) = 1 · K G ( m − n − o 1 , − o 2 , . . . , − o n , 0) .

Lidskii volume formula Theorem (Baldoni-Vergne 08, Postnikov-Stanley - unpublished) G m edges, n + 1 vertices, a i ≥ 0 � m − n � � a j 1 1 · · · a j n vol F G ( a 1 , . . . , a n ) = n j 1 , . . . , j n j � o × K G ( j 1 − o 1 , . . . , j n − o n , 0) where o = ( o 1 , . . . , o n ) , o v = outdeg ( v ) − 1 and | j | = m − n . Corollary: vol F G (1 , 0 , . . . , 0 , − 1) = 1 · K G ( m − n − o 1 , − o 2 , . . . , − o n , 0) . Example: (CRY polytope) � n − 1 � vol F k n +1 (1 , 0 , . . . , 0 , − 1) = K k n +1 ( , − n + 2 , . . . , − 2 , − 1 , 0) 2

Lidskii lattice point formula Theorem (Baldoni-Vergne 08, Postnikov-Stanley – unpublished) G m edges, n + 1 vertices, a i ≥ 0 � � a 1 − i 1 � � � � a n − i n � � � K G ( a 1 , . . . , a n ) = · · · j 1 j n j � o × K G ( j 1 − o 1 , . . . , j n − o n , 0) where | j | = m − n , o v = outdeg ( v ) − 1 , i v = indeg ( v ) − 1

Lidskii lattice point formula Theorem (Baldoni-Vergne 08, Postnikov-Stanley – unpublished) G m edges, n + 1 vertices, a i ≥ 0 � � a 1 − i 1 � � � � a n − i n � � � K G ( a 1 , . . . , a n ) = · · · j 1 j n j � o × K G ( j 1 − o 1 , . . . , j n − o n , 0) where | j | = m − n , o v = outdeg ( v ) − 1 , i v = indeg ( v ) − 1 Pitman-Stanley polytope: Π 4 � � a 1 + 1 � � � � a 2 � � � � a n � � � F Π n +1 ( a ) = · · · j 1 j 2 j n j � (1 ,..., 1)

Lidskii lattice point formula Theorem (Baldoni-Vergne 08, Postnikov-Stanley – unpublished) G m edges, n + 1 vertices, a i ≥ 0 � � a 1 − i 1 � � � � a n − i n � � � K G ( a 1 , . . . , a n ) = · · · j 1 j n j � o × K G ( j 1 − o 1 , . . . , j n − o n , 0) where | j | = m − n , o v = outdeg ( v ) − 1 , i v = indeg ( v ) − 1 Example K G (1 , 1 , − 2) = � � 2 � � � � 0 � � � � 2 � � � � 0 � � = K G (0 , 0 , 0) + K G (1 , − 1 , 0) 1 1 2 0 = 0 + 3 · 2 = 6 .

About the proofs � � a 1 − i 1 � � � � a n − i n � � � K G ( a 1 , . . . , a n ) = · · · j 1 j n j � o × K G ( j 1 − o 1 , . . . , j n − o n , 0) • proof by Baldoni and Vergne uses residues • proof by Postnikov-Stanley uses the Elliott-MacMahon algorithm

About the proofs � � a 1 − i 1 � � � � a n − i n � � � K G ( a 1 , . . . , a n ) = · · · j 1 j n j � o × K G ( j 1 − o 1 , . . . , j n − o n , 0) • proof by Baldoni and Vergne uses residues • proof by Postnikov-Stanley uses the Elliott-MacMahon algorithm • new proof (M-Morales) by polytope subdivision

Subdivision proof of Lidskii formulas � m − n � � a j 1 1 · · · a j n vol F G ( a 1 , . . . , a n ) = n j 1 , . . . , j n j � o × K G ( j 1 − o 1 , . . . , j n − o n , 0) Subdivide F G ( a ) into cells of types indexed by j . j 1 + 1 multiple edges j 2 + 1 multiple edges a 1 a 2 a 3 � m − n � a j 1 1 · · · a j n volume of each type j cell : n j 1 , . . . , j n # times type j cell appears: K G ( j 1 − o 1 , . . . , j n − o n , 0)

Example subdivision 1 1 − 2

Example subdivision 1 1 − 2 × lower dimen- sional

Example subdivision 1 1 − 2 × lower dimen- sional

Example subdivision 1 1 − 2 × lower dimen- sional

Example subdivision 1 1 − 2 volume: 2 · 1+1 · 2 = 4 . × lower lattice points: dimen- 0 + 3 · 2 = 6 . sional

Triangulation in the case F G (1 , 0 , . . . , 0 , − 1) Special case (Stanley-Postnikov) vol F G (1 , 0 , . . . , 0 , − 1) = 1 · K G ( m − n − o 1 , − o 2 , . . . , − o n , 0) .

Triangulation in the case F G (1 , 0 , . . . , 0 , − 1) Special case (Stanley-Postnikov) vol F G (1 , 0 , . . . , 0 , − 1) = 1 · K G ( m − n − o 1 , − o 2 , . . . , − o n , 0) . Final outcomes of subdivision all of the form m − n + 1 multiple edges 1 − 1

Triangulation in the case F G (1 , 0 , . . . , 0 , − 1) Special case (Stanley-Postnikov) vol F G (1 , 0 , . . . , 0 , − 1) = 1 · K G ( m − n − o 1 , − o 2 , . . . , − o n , 0) . Final outcomes of subdivision all of the form m − n + 1 multiple edges 1 − 1 • the associated polytope is a simplex with volume 1

Triangulation in the case F G (1 , 0 , . . . , 0 , − 1) Special case (Stanley-Postnikov) vol F G (1 , 0 , . . . , 0 , − 1) = 1 · K G ( m − n − o 1 , − o 2 , . . . , − o n , 0) . Final outcomes of subdivision all of the form m − n + 1 multiple edges 1 − 1 • the associated polytope is a simplex with volume 1 • They proved the number of times we obtain this outcome in the subdivision tree is K G ( m − n − o 1 , − o 2 , . . . , − o n , 0)

Flow polytopes and...

Flow polytopes and... • Kostant partition functions

Flow polytopes and... • Kostant partition functions

Flow polytopes and... • Kostant partition functions • Grothendieck polynomials

Flow polytopes and... • Kostant partition functions • Grothendieck polynomials • space of diagonal harmonics

Encoding triangulations of F ( G ) := F G (1 , 0 , . . . , 0 , − 1) G 0 G 1 G 2 G 3 → Reduction Lemma. F ( G 0 ) = F ( G 1 ) ∪ F ( G 2 ) , F ( G 1 ) ∩ F ( G 2 ) = F ( G 3 ) .

Encoding triangulations of F ( G ) := F G (1 , 0 , . . . , 0 , − 1) G 0 G 1 G 2 G 3 → Reduction Lemma. F ( G 0 ) = F ( G 1 ) ∪ F ( G 2 ) , F ( G 1 ) ∩ F ( G 2 ) = F ( G 3 ) . Some of the polytopes above may be empty.

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� G s t

� G s t

� G s t

Encoding triangulations of F ( � G ) G 0 G 1 G 2 G 3 → Reduction Lemma. F ( � G 0 ) = F ( � G 1 ) ∪ F ( � G 2 ) , F ( � G 1 ) ∩ F ( � G 2 ) = F ( � G 3 ) , where F ( � G 0 ) , F ( � G 1 ) , F ( � G 2 ) , are of the same dimension and F ( � G 3 ) is one dimension less.

Reduction tree T ( G ) 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 A reduction tree of G = ([4] , { (1 , 2) , (2 , 3) , (3 , 4) } ) with five leaves. The edges on which the reductions are performed are in bold.

Reduction tree T ( G ) 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 If the leaves are labeled by graphs H 1 , . . . , H k then the flow polytopes F ( � H 1 ) , . . . , F ( � H k ) are simplices.

Canonical reduction tree

Canonical reduction tree

Canonical reduction tree Theorem (M, 2009) The full dimensional leaves of the canonical reduction tree are the noncrossing alternating spanning trees of the directed transitive closure of the noncrossing tree at the root.

Encoding triangulations with reduced forms G 0 G 1 G 2 G 3 → i j k i j i j k i j k k x ij x jk → x jk x ik + x ik x ij + βx ik

The subdivision algebra Generated by x ij , 1 ≤ i < j ≤ n , over Z [ β ] subject to relations x ij x kl = x kl x ij , for all i, j, k, l x ij x jk = x jk x ik + x ik x ij + βx ik