Hodge theory in combinatorics Eric Katz (University of Waterloo) joint with June Huh (IAS) and Karim Adiprasito (IAS) May 14, 2015 “But Hodge shan’t be shot; no, no, Hodge shall not be shot.” – Samuel Johnson Eric Katz (Waterloo) HTIC May 14, 2015 1 / 30
The characteristic polynomial of a subspace Let k be a field. Let V ⊂ k n +1 be an ( r + 1)-dim linear subspace not contained in any coordinate hyperplane. Would like to use inclusion/exclusion to express [ V ∩ ( k ∗ ) n +1 ] as a linear combination of [ V ∩ L I ]’s where L I is the coordinate subspace given by L I = { x i 1 = x i 2 = · · · = x i l = 0 } for I = { i 1 , i 2 , . . . , i l } ⊂ { 0 , . . . , n } . Eric Katz (Waterloo) HTIC May 14, 2015 2 / 30
The characteristic polynomial of a subspace Let k be a field. Let V ⊂ k n +1 be an ( r + 1)-dim linear subspace not contained in any coordinate hyperplane. Would like to use inclusion/exclusion to express [ V ∩ ( k ∗ ) n +1 ] as a linear combination of [ V ∩ L I ]’s where L I is the coordinate subspace given by L I = { x i 1 = x i 2 = · · · = x i l = 0 } for I = { i 1 , i 2 , . . . , i l } ⊂ { 0 , . . . , n } . Example: Let V be a generic subspace (intersecting every coordinate subspace in the expected dimension). Then � � � [ V ∩ (( k ∗ ) n +1 )] = [ V ∩ L ∅ ] − [ V ∩ L i ] + [ V ∩ L I ] − [ V ∩ L I ] + . . . . i I I | I | =2 | I | =3 Eric Katz (Waterloo) HTIC May 14, 2015 2 / 30
The characteristic polynomial of a subspace Let k be a field. Let V ⊂ k n +1 be an ( r + 1)-dim linear subspace not contained in any coordinate hyperplane. Would like to use inclusion/exclusion to express [ V ∩ ( k ∗ ) n +1 ] as a linear combination of [ V ∩ L I ]’s where L I is the coordinate subspace given by L I = { x i 1 = x i 2 = · · · = x i l = 0 } for I = { i 1 , i 2 , . . . , i l } ⊂ { 0 , . . . , n } . Example: Let V be a generic subspace (intersecting every coordinate subspace in the expected dimension). Then � � � [ V ∩ (( k ∗ ) n +1 )] = [ V ∩ L ∅ ] − [ V ∩ L i ] + [ V ∩ L I ] − [ V ∩ L I ] + . . . . i I I | I | =2 | I | =3 If you’re fancy, you can say that this is a motivic expression. Eric Katz (Waterloo) HTIC May 14, 2015 2 / 30
Flats In general, you may have to be a little more careful as there may be I , J ⊆ { 0 , . . . , n } with V ∩ L I = V ∩ L J . Need to make sure we do not overcount. Eric Katz (Waterloo) HTIC May 14, 2015 3 / 30
Flats In general, you may have to be a little more careful as there may be I , J ⊆ { 0 , . . . , n } with V ∩ L I = V ∩ L J . Need to make sure we do not overcount. Definition A subset I ⊂ { 0 , . . . , n } is said to be a flat if for any J ⊃ I , V ∩ L J � = V ∩ L I . Eric Katz (Waterloo) HTIC May 14, 2015 3 / 30
Flats In general, you may have to be a little more careful as there may be I , J ⊆ { 0 , . . . , n } with V ∩ L I = V ∩ L J . Need to make sure we do not overcount. Definition A subset I ⊂ { 0 , . . . , n } is said to be a flat if for any J ⊃ I , V ∩ L J � = V ∩ L I . The rank of a flat is ρ ( I ) = codim( V ∩ L I ⊂ V ) . Eric Katz (Waterloo) HTIC May 14, 2015 3 / 30
Flats In general, you may have to be a little more careful as there may be I , J ⊆ { 0 , . . . , n } with V ∩ L I = V ∩ L J . Need to make sure we do not overcount. Definition A subset I ⊂ { 0 , . . . , n } is said to be a flat if for any J ⊃ I , V ∩ L J � = V ∩ L I . The rank of a flat is ρ ( I ) = codim( V ∩ L I ⊂ V ) . We can now write for some choice of ν I ∈ Z , � [ V ∩ ( k ∗ ) n +1 ] = ν I [ V ∩ L I ] . flats I Eric Katz (Waterloo) HTIC May 14, 2015 3 / 30
Flats In general, you may have to be a little more careful as there may be I , J ⊆ { 0 , . . . , n } with V ∩ L I = V ∩ L J . Need to make sure we do not overcount. Definition A subset I ⊂ { 0 , . . . , n } is said to be a flat if for any J ⊃ I , V ∩ L J � = V ∩ L I . The rank of a flat is ρ ( I ) = codim( V ∩ L I ⊂ V ) . We can now write for some choice of ν I ∈ Z , � [ V ∩ ( k ∗ ) n +1 ] = ν I [ V ∩ L I ] . flats I Fact: ( − 1) ρ ( I ) ν V is always positive. Eric Katz (Waterloo) HTIC May 14, 2015 3 / 30
Characteristic Polynomial Definition The characteristic polynomial of V is r +1 � � q r +1 − i χ V ( q ) = ν I i =0 flats I ρ ( I )= i µ 0 q r +1 − µ 1 q r + · · · + ( − 1) r +1 µ r +1 ≡ Eric Katz (Waterloo) HTIC May 14, 2015 4 / 30
Characteristic Polynomial Definition The characteristic polynomial of V is r +1 � � q r +1 − i χ V ( q ) = ν I i =0 flats I ρ ( I )= i µ 0 q r +1 − µ 1 q r + · · · + ( − 1) r +1 µ r +1 ≡ We can think of χ as an evaluation of the classes [ V ∩ L I ] of the form [ V ∩ L I ] �→ q r +1 − ρ ( I ) so the characteristic polynomial is the image of [ V ∩ ( k ∗ ) n +1 ]. Eric Katz (Waterloo) HTIC May 14, 2015 4 / 30
Characteristic Polynomial Definition The characteristic polynomial of V is r +1 � � q r +1 − i χ V ( q ) = ν I i =0 flats I ρ ( I )= i µ 0 q r +1 − µ 1 q r + · · · + ( − 1) r +1 µ r +1 ≡ We can think of χ as an evaluation of the classes [ V ∩ L I ] of the form [ V ∩ L I ] �→ q r +1 − ρ ( I ) so the characteristic polynomial is the image of [ V ∩ ( k ∗ ) n +1 ]. Example: In the generic case subspace case, we have � r + 1 � � r + 1 � � r + 1 � χ V ( q ) = q r +1 − q r + q r − 1 − · · · + ( − 1) r +1 . 1 2 r + 1 Eric Katz (Waterloo) HTIC May 14, 2015 4 / 30
Rota-Heron-Welsh Conjecture Theorem (Rota-Heron-Welsh Conjecture (in the realizable case) (Huh-k ’11)) χ V ( q ) is log-concave and internal zero-free, hence unimodal. Eric Katz (Waterloo) HTIC May 14, 2015 5 / 30
Rota-Heron-Welsh Conjecture Theorem (Rota-Heron-Welsh Conjecture (in the realizable case) (Huh-k ’11)) χ V ( q ) is log-concave and internal zero-free, hence unimodal. Definition A polynomial with coefficients µ 0 , . . . , µ r +1 is said to be log-concave if for all i , | µ i − 1 µ i +1 | ≤ µ 2 i . (so log of coefficients is a concave sequence.) Eric Katz (Waterloo) HTIC May 14, 2015 5 / 30
Rota-Heron-Welsh Conjecture Theorem (Rota-Heron-Welsh Conjecture (in the realizable case) (Huh-k ’11)) χ V ( q ) is log-concave and internal zero-free, hence unimodal. Definition A polynomial with coefficients µ 0 , . . . , µ r +1 is said to be log-concave if for all i , | µ i − 1 µ i +1 | ≤ µ 2 i . (so log of coefficients is a concave sequence.) Definition A polynomial with coefficients µ 0 , . . . , µ r +1 is said to be unimodal if the coefficients are unimodal in absolute value, i.e. there is a j such that | µ 0 | ≤ | µ 1 | ≤ · · · ≤ | µ j | ≥ | µ j +1 | ≥ · · · ≥ | µ r +1 | . Eric Katz (Waterloo) HTIC May 14, 2015 5 / 30
Motivation:Chromatic Polynomials of Graphs Original Motivation: Let Γ be a loop-free graph. Define the chromatic function χ Γ by setting χ Γ ( q ) to be the number of colorings of Γ with q colors such that no edge connects vertices of the same color. Eric Katz (Waterloo) HTIC May 14, 2015 6 / 30
Motivation:Chromatic Polynomials of Graphs Original Motivation: Let Γ be a loop-free graph. Define the chromatic function χ Γ by setting χ Γ ( q ) to be the number of colorings of Γ with q colors such that no edge connects vertices of the same color. Fact: χ Γ ( q ) is a polynomial of degree equal to the number of vertices with alternating coefficients. Eric Katz (Waterloo) HTIC May 14, 2015 6 / 30
Motivation:Chromatic Polynomials of Graphs Original Motivation: Let Γ be a loop-free graph. Define the chromatic function χ Γ by setting χ Γ ( q ) to be the number of colorings of Γ with q colors such that no edge connects vertices of the same color. Fact: χ Γ ( q ) is a polynomial of degree equal to the number of vertices with alternating coefficients. Read’s Conjecture ’68 (Huh ’10): χ Γ ( q ) is unimodal. Eric Katz (Waterloo) HTIC May 14, 2015 6 / 30
Matroids We may abstract the linear space to a rank function ρ : 2 { 0 ,..., n } → Z satisfying Eric Katz (Waterloo) HTIC May 14, 2015 7 / 30
Matroids We may abstract the linear space to a rank function ρ : 2 { 0 ,..., n } → Z satisfying 1 0 ≤ ρ ( I ) ≤ | I | Eric Katz (Waterloo) HTIC May 14, 2015 7 / 30
Matroids We may abstract the linear space to a rank function ρ : 2 { 0 ,..., n } → Z satisfying 1 0 ≤ ρ ( I ) ≤ | I | 2 I ⊂ J implies ρ ( I ) ≤ ρ ( J ) Eric Katz (Waterloo) HTIC May 14, 2015 7 / 30
Matroids We may abstract the linear space to a rank function ρ : 2 { 0 ,..., n } → Z satisfying 1 0 ≤ ρ ( I ) ≤ | I | 2 I ⊂ J implies ρ ( I ) ≤ ρ ( J ) 3 ρ ( I ∪ J ) + ρ ( I ∩ J ) ≤ ρ ( I ) + ρ ( J ) Eric Katz (Waterloo) HTIC May 14, 2015 7 / 30
Matroids We may abstract the linear space to a rank function ρ : 2 { 0 ,..., n } → Z satisfying 1 0 ≤ ρ ( I ) ≤ | I | 2 I ⊂ J implies ρ ( I ) ≤ ρ ( J ) 3 ρ ( I ∪ J ) + ρ ( I ∩ J ) ≤ ρ ( I ) + ρ ( J ) 4 ρ ( { 0 , . . . , n } ) = r + 1 . Eric Katz (Waterloo) HTIC May 14, 2015 7 / 30
Recommend
More recommend
