Deciding Positivity of Kronecker Coefficients is NP-hard
Christian Ikenmeyer
Texas A&M University
joint work with Ketan Mulmuley
University of Chicago
and Michael Walter
Stanford University
SIAM Conference on Applied Algebraic Geometry 2015
Christian Ikenmeyer
Deciding Positivity of Kronecker Coefficients is NP-hard
1
Motivation: Kronecker coefficients and tensors
The m × m matrix multiplication tensor Mm lives in the space
Christian Ikenmeyer
N3
2
Cm .
Deciding Positivity of Kronecker Coefficients is NP-hard
2
Motivation: Kronecker coefficients and tensors
N
2
The m × m matrix multiplication tensor Mm lives in the space 3 Cm .
N
2
Let Xn ⊆ 3 Cm denote the subvariety of border rank ≤ n tensors.
Christian Ikenmeyer
Deciding Positivity of Kronecker Coefficients is NP-hard
2
Motivation: Kronecker coefficients and tensors
N
2
The m × m matrix multiplication tensor Mm lives in the space 3 Cm .
N
2
Let Xn ⊆ 3 Cm denote the subvariety of border rank ≤ n tensors.
N
2
To prove Mm ∈
/ Xn we study the vanishing ideal I (Xn ) ⊆ O( 3 Cm ), because we
want to find f ∈ I (Xn ) such that f (Mm ) 6= 0.
Christian Ikenmeyer
Deciding Positivity of Kronecker Coefficients is NP-hard
2
Motivation: Kronecker coefficients and tensors
N
2
The m × m matrix multiplication tensor Mm lives in the space 3 Cm .
N
2
Let Xn ⊆ 3 Cm denote the subvariety of border rank ≤ n tensors.
N
2
To prove Mm ∈
/ Xn we study the vanishing ideal I (Xn ) ⊆ O( 3 Cm ), because we
want to find f ∈ I (Xn ) such that f (Mm ) 6= 0.
N
2
The group GL3m2 := GLm2 × GLm2 × GLm2 acts canonically on 3 Cm .
Christian Ikenmeyer
Deciding Positivity of Kronecker Coefficients is NP-hard
2
Motivation: Kronecker coefficients and tensors
N
2
The m × m matrix multiplication tensor Mm lives in the space 3 Cm .
N
2
Let Xn ⊆ 3 Cm denote the subvariety of border rank ≤ n tensors.
N
2
To prove Mm ∈
/ Xn we study the vanishing ideal I (Xn ) ⊆ O( 3 Cm ), because we
want to find f ∈ I (Xn ) such that f (Mm ) 6= 0.
N
2
The group GL3m2 := GLm2 × GLm2 × GLm2 acts canonically on 3 Cm .
N3 m 2
For every degree d: O(
C )d is a fin. dim. GL3m2 -representation and I (Xn )d is a
subrepresentation.
Christian Ikenmeyer
Deciding Positivity of Kronecker Coefficients is NP-hard
2
Motivation: Kronecker coefficients and tensors
N
2
The m × m matrix multiplication tensor Mm lives in the space 3 Cm .
N
2
Let Xn ⊆ 3 Cm denote the subvariety of border rank ≤ n tensors.
N
2
To prove Mm ∈
/ Xn we study the vanishing ideal I (Xn ) ⊆ O( 3 Cm ), because we
want to find f ∈ I (Xn ) such that f (Mm ) 6= 0.
N
2
The group GL3m2 := GLm2 × GLm2 × GLm2 acts canonically on 3 Cm .
N3 m 2
For every degree d: O(
C )d is a fin. dim. GL3m2 -representation and I (Xn )d is a
N
2
subrepresentation. Since GL3m2 is reductive, both O( 3 Cm )d and I (Xn )d can be
decomposed into sums of irreducible representations.
Christian Ikenmeyer
Deciding Positivity of Kronecker Coefficients is NP-hard
2
Motivation: Kronecker coefficients and tensors
N
2
The m × m matrix multiplication tensor Mm lives in the space 3 Cm .
N
2
Let Xn ⊆ 3 Cm denote the subvariety of border rank ≤ n tensors.
N
2
To prove Mm ∈
/ Xn we study the vanishing ideal I (Xn ) ⊆ O( 3 Cm ), because we
want to find f ∈ I (Xn ) such that f (Mm ) 6= 0.
N
2
The group GL3m2 := GLm2 × GLm2 × GLm2 acts canonically on 3 Cm .
N3 m 2
For every degree d: O(
C )d is a fin. dim. GL3m2 -representation and I (Xn )d is a
N
2
subrepresentation. Since GL3m2 is reductive, both O( 3 Cm )d and I (Xn )d can be
decomposed into sums of irreducible representations. We can restrict our search to
those f that lie in an irreducible representation.
Christian Ikenmeyer
Deciding Positivity of Kronecker Coefficients is NP-hard
2
Motivation: Kronecker coefficients and tensors
N
2
The m × m matrix multiplication tensor Mm lives in the space 3 Cm .
N
2
Let Xn ⊆ 3 Cm denote the subvariety of border rank ≤ n tensors.
N
2
To prove Mm ∈
/ Xn we study the vanishing ideal I (Xn ) ⊆ O( 3 Cm ), because we
want to find f ∈ I (Xn ) such that f (Mm ) 6= 0.
N
2
The group GL3m2 := GLm2 × GLm2 × GLm2 acts canonically on 3 Cm .
N3 m 2
For every degree d: O(
C )d is a fin. dim. GL3m2 -representation and I (Xn )d is a
N
2
subrepresentation. Since GL3m2 is reductive, both O( 3 Cm )d and I (Xn )d can be
decomposed into sums of irreducible representations. We can restrict our search to
those f that lie in an irreducible representation.
Representation theory: Polynomial irreducible GL3m2 -representations are indexed by
triples of partitions: E.g. ((5, 1, 1, 1), (2, 2, 2, 2), (2, 2, 2, 2)) is a triple of partitions
of 8.
Notation for an irreducible GL3m2 -representation: Vλ,µ,ν .
Christian Ikenmeyer
Deciding Positivity of Kronecker Coefficients is NP-hard
2
Motivation: Kronecker coefficients and tensors
N
2
The m × m matrix multiplication tensor Mm lives in the space 3 Cm .
N
2
Let Xn ⊆ 3 Cm denote the subvariety of border rank ≤ n tensors.
N
2
To prove Mm ∈
/ Xn we study the vanishing ideal I (Xn ) ⊆ O( 3 Cm ), because we
want to find f ∈ I (Xn ) such that f (Mm ) 6= 0.
N
2
The group GL3m2 := GLm2 × GLm2 × GLm2 acts canonically on 3 Cm .
N3 m 2
For every degree d: O(
C )d is a fin. dim. GL3m2 -representation and I (Xn )d is a
N
2
subrepresentation. Since GL3m2 is reductive, both O( 3 Cm )d and I (Xn )d can be
decomposed into sums of irreducible representations. We can restrict our search to
those f that lie in an irreducible representation.
Representation theory: Polynomial irreducible GL3m2 -representations are indexed by
triples of partitions: E.g. ((5, 1, 1, 1), (2, 2, 2, 2), (2, 2, 2, 2)) is a triple of partitions
of 8.
Notation for an irreducible GL3m2 -representation: Vλ,µ,ν .
N
2
The representation theoretic decomposition of O( 3 Cm ) is given by Kronecker
coefficients gλ,µ,ν :
M
2
N
O( 3 Cm )d =
gλ,µ,ν Vλ,µ,ν
(λ,µ,ν)
where the sum if over triples of partitions with d boxes and at most m2 parts.
Christian Ikenmeyer
Deciding Positivity of Kronecker Coefficients is NP-hard
2
Motivation: Kronecker coefficients and matrix multiplication
Moreover, quite surprisingly the same coefficients arise in a different question:
N
2
Recall that GL3m2 acts on 3 Cm and consider the orbit
3
3
GLm2 Mm := {gMm | g ∈ GLm2 }.
The representation theory of its coordinate ring is given by (Bürgisser, I. 2010):
M X
O(GL3m2 Mm )d =
g (λ̃, µ, ν)g (λ, µ̃, ν)g (λ, µ, ν̃) Vλ,µ,ν
(λ,µ,ν)`d
(λ̃,µ̃,ν̃)`d
where the outer sum is over triples of partition with d boxes and at most m2 parts
and the inner sum is over triples of partitions with d boxes and at most m parts.
Christian Ikenmeyer
Deciding Positivity of Kronecker Coefficients is NP-hard
3
Hardness of Kronecker Coefficients
Kronecker coefficients gλ,µ,ν are #P-hard to compute (2008).
Christian Ikenmeyer
Deciding Positivity of Kronecker Coefficients is NP-hard
4
Hardness of Kronecker Coefficients
Kronecker coefficients gλ,µ,ν are #P-hard to compute (2008).
A nontrivial subcase of Kronecker coefficients are the well known
Littlewood-Richardson coefficients and even those are #P-hard to compute. But
positivity can be decided in polynomial time.
Christian Ikenmeyer
Deciding Positivity of Kronecker Coefficients is NP-hard
4
Hardness of Kronecker Coefficients
Kronecker coefficients gλ,µ,ν are #P-hard to compute (2008).
A nontrivial subcase of Kronecker coefficients are the well known
Littlewood-Richardson coefficients and even those are #P-hard to compute. But
positivity can be decided in polynomial time.
Main tool for Littlewood-Richardson coefficients: A combinatorial description, i.e.,
their computation is in #P.
Christian Ikenmeyer
Deciding Positivity of Kronecker Coefficients is NP-hard
4
Hardness of Kronecker Coefficients
Kronecker coefficients gλ,µ,ν are #P-hard to compute (2008).
A nontrivial subcase of Kronecker coefficients are the well known
Littlewood-Richardson coefficients and even those are #P-hard to compute. But
positivity can be decided in polynomial time.
Main tool for Littlewood-Richardson coefficients: A combinatorial description, i.e.,
their computation is in #P.
No combinatorial description is known for Kronecker coefficients.
Christian Ikenmeyer
Deciding Positivity of Kronecker Coefficients is NP-hard
4
Hardness of Kronecker Coefficients
Kronecker coefficients gλ,µ,ν are #P-hard to compute (2008).
A nontrivial subcase of Kronecker coefficients are the well known
Littlewood-Richardson coefficients and even those are #P-hard to compute. But
positivity can be decided in polynomial time.
Main tool for Littlewood-Richardson coefficients: A combinatorial description, i.e.,
their computation is in #P.
No combinatorial description is known for Kronecker coefficients.
The hope was that positivity of Kronecker coefficients could also be decided in
polynomial time (conjectured be Mulmuley).
Christian Ikenmeyer
Deciding Positivity of Kronecker Coefficients is NP-hard
4
Hardness of Kronecker Coefficients
Kronecker coefficients gλ,µ,ν are #P-hard to compute (2008).
A nontrivial subcase of Kronecker coefficients are the well known
Littlewood-Richardson coefficients and even those are #P-hard to compute. But
positivity can be decided in polynomial time.
Main tool for Littlewood-Richardson coefficients: A combinatorial description, i.e.,
their computation is in #P.
No combinatorial description is known for Kronecker coefficients.
The hope was that positivity of Kronecker coefficients could also be decided in
polynomial time (conjectured be Mulmuley).
Theorem (IMW 2015)
Deciding positivity of Kronecker coefficients is NP-hard.
Christian Ikenmeyer
Deciding Positivity of Kronecker Coefficients is NP-hard
4
Hardness of Kronecker Coefficients
Kronecker coefficients gλ,µ,ν are #P-hard to compute (2008).
A nontrivial subcase of Kronecker coefficients are the well known
Littlewood-Richardson coefficients and even those are #P-hard to compute. But
positivity can be decided in polynomial time.
Main tool for Littlewood-Richardson coefficients: A combinatorial description, i.e.,
their computation is in #P.
No combinatorial description is known for Kronecker coefficients.
The hope was that positivity of Kronecker coefficients could also be decided in
polynomial time (conjectured be Mulmuley).
Theorem (IMW 2015)
Deciding positivity of Kronecker coefficients is NP-hard.
Proof idea: There is an upper bound and a lower bound
pλ,µ,ν ≤ gλ,µ,ν ≤ tλ,µ,ν
and both bounds are combinatorial, coming from discrete tomography.
Christian Ikenmeyer
Deciding Positivity of Kronecker Coefficients is NP-hard
4
Hardness of Kronecker Coefficients
Kronecker coefficients gλ,µ,ν are #P-hard to compute (2008).
A nontrivial subcase of Kronecker coefficients are the well known
Littlewood-Richardson coefficients and even those are #P-hard to compute. But
positivity can be decided in polynomial time.
Main tool for Littlewood-Richardson coefficients: A combinatorial description, i.e.,
their computation is in #P.
No combinatorial description is known for Kronecker coefficients.
The hope was that positivity of Kronecker coefficients could also be decided in
polynomial time (conjectured be Mulmuley).
Theorem (IMW 2015)
Deciding positivity of Kronecker coefficients is NP-hard.
Proof idea: There is an upper bound and a lower bound
pλ,µ,ν ≤ gλ,µ,ν ≤ tλ,µ,ν
and both bounds are combinatorial, coming from discrete tomography.
In some cases we have equality pλ,µ,ν = gλ,µ,ν = tλ,µ,ν . In these cases deciding
positivity is NP-hard.
Christian Ikenmeyer
Deciding Positivity of Kronecker Coefficients is NP-hard
4
The upper bound tλ,µ,ν is defined as the number of point sets that have slice sizes
λt in x-direction, µt in y -direction, and ν t in z-direction.
In this example the slice sizes are: (3,2,1,0), (4,1,1,0), (5,1,0,0)
Christian Ikenmeyer
Deciding Positivity of Kronecker Coefficients is NP-hard
5
The upper bound tλ,µ,ν is defined as the number of point sets that have slice sizes
λt in x-direction, µt in y -direction, and ν t in z-direction.
In this example the slice sizes are: (3,2,1,0), (4,1,1,0), (5,1,0,0)
Christian Ikenmeyer
Deciding Positivity of Kronecker Coefficients is NP-hard
5
The upper bound tλ,µ,ν is defined as the number of point sets that have slice sizes
λt in x-direction, µt in y -direction, and ν t in z-direction.
In this example the slice sizes are: (3,2,1,0), (4,1,1,0), (5,1,0,0)
Christian Ikenmeyer
Deciding Positivity of Kronecker Coefficients is NP-hard
5
The upper bound tλ,µ,ν is defined as the number of point sets that have slice sizes
λt in x-direction, µt in y -direction, and ν t in z-direction.
In this example the slice sizes are: (3,2,1,0), (4,1,1,0), (5,1,0,0)
Christian Ikenmeyer
Deciding Positivity of Kronecker Coefficients is NP-hard
5
The upper bound tλ,µ,ν is defined as the number of point sets that have slice sizes
λt in x-direction, µt in y -direction, and ν t in z-direction.
In this example the slice sizes are: (3,2,1,0), (4,1,1,0), (5,1,0,0)
Christian Ikenmeyer
Deciding Positivity of Kronecker Coefficients is NP-hard
5
The upper bound tλ,µ,ν is defined as the number of point sets that have slice sizes
λt in x-direction, µt in y -direction, and ν t in z-direction.
In this example the slice sizes are: (3,2,1,0), (4,1,1,0), (5,1,0,0)
Christian Ikenmeyer
Deciding Positivity of Kronecker Coefficients is NP-hard
5
The upper bound tλ,µ,ν is defined as the number of point sets that have slice sizes
λt in x-direction, µt in y -direction, and ν t in z-direction.
In this example the slice sizes are: (3,2,1,0), (4,1,1,0), (5,1,0,0)
Christian Ikenmeyer
Deciding Positivity of Kronecker Coefficients is NP-hard
5
The upper bound tλ,µ,ν is defined as the number of point sets that have slice sizes
λt in x-direction, µt in y -direction, and ν t in z-direction.
In this example the slice sizes are: (3,2,1,0), (4,1,1,0), (5,1,0,0)
Christian Ikenmeyer
Deciding Positivity of Kronecker Coefficients is NP-hard
5
The upper bound tλ,µ,ν is defined as the number of point sets that have slice sizes
λt in x-direction, µt in y -direction, and ν t in z-direction.
In this example the slice sizes are: (3,2,1,0), (4,1,1,0), (5,1,0,0)
Christian Ikenmeyer
Deciding Positivity of Kronecker Coefficients is NP-hard
5
The upper bound tλ,µ,ν is defined as the number of point sets that have slice sizes
λt in x-direction, µt in y -direction, and ν t in z-direction.
In this example the slice sizes are: (3,2,1,0), (4,1,1,0), (5,1,0,0)
Christian Ikenmeyer
Deciding Positivity of Kronecker Coefficients is NP-hard
5
The upper bound tλ,µ,ν is defined as the number of point sets that have slice sizes
λt in x-direction, µt in y -direction, and ν t in z-direction.
In this example the slice sizes are: (3,2,1,0), (4,1,1,0), (5,1,0,0)
Christian Ikenmeyer
Deciding Positivity of Kronecker Coefficients is NP-hard
5
The upper bound tλ,µ,ν is defined as the number of point sets that have slice sizes
λt in x-direction, µt in y -direction, and ν t in z-direction.
In this example the slice sizes are: (3,2,1,0), (4,1,1,0), (5,1,0,0)
Christian Ikenmeyer
Deciding Positivity of Kronecker Coefficients is NP-hard
5
The upper bound tλ,µ,ν is defined as the number of point sets that have slice sizes
λt in x-direction, µt in y -direction, and ν t in z-direction.
In this example the slice sizes are: (3,2,1,0), (4,1,1,0), (5,1,0,0)
Christian Ikenmeyer
Deciding Positivity of Kronecker Coefficients is NP-hard
5
The upper bound tλ,µ,ν is defined as the number of point sets that have slice sizes
λt , µ t , ν t .
Q is a pyramid if: (x, y , z) ∈ Q ⇒ (x 0 , y 0 , z 0 ) ∈ Q for all x 0 ≤ x, y 0 ≤ y , z 0 ≤ z.
The lower bound pλ,µ,ν is defined as the number of pyramids that have slice sizes
λt , µt , ν t (Manivel 1997).
pλ,µ,ν ≤ gλ,µ,ν ≤ tλ,µ,ν .
Christian Ikenmeyer
Deciding Positivity of Kronecker Coefficients is NP-hard
6
Simplex-like margin triples
The barycenter norm
three margin vectors.
Christian Ikenmeyer
P
q∈Q (qx
+ qy + qz ) of a point set Q depends only on the
Deciding Positivity of Kronecker Coefficients is NP-hard
7
Simplex-like margin triples
The barycenter norm
three margin vectors.
P
q∈Q (qx
+ qy + qz ) of a point set Q depends only on the
If we have given margins with minimal barycenter norm, then to construct Q with
these margins we must place everything as close to the origin as possible (in taxicab
distance).
Christian Ikenmeyer
Deciding Positivity of Kronecker Coefficients is NP-hard
7
Simplex-like margin triples
The barycenter norm
three margin vectors.
P
q∈Q (qx
+ qy + qz ) of a point set Q depends only on the
If we have given margins with minimal barycenter norm, then to construct Q with
these margins we must place everything as close to the origin as possible (in taxicab
distance).
Margin triples with minimal barycenter norm are called simplex-like.
Christian Ikenmeyer
Deciding Positivity of Kronecker Coefficients is NP-hard
7
Simplex-like margin triples
The barycenter norm
three margin vectors.
P
q∈Q (qx
+ qy + qz ) of a point set Q depends only on the
If we have given margins with minimal barycenter norm, then to construct Q with
these margins we must place everything as close to the origin as possible (in taxicab
distance).
Margin triples with minimal barycenter norm are called simplex-like.
Point sets with simplex-like margin triples are completely filled in all but the last
layer.
Christian Ikenmeyer
Deciding Positivity of Kronecker Coefficients is NP-hard
7
Simplex-like margin triples
The barycenter norm
three margin vectors.
P
q∈Q (qx
+ qy + qz ) of a point set Q depends only on the
If we have given margins with minimal barycenter norm, then to construct Q with
these margins we must place everything as close to the origin as possible (in taxicab
distance).
Margin triples with minimal barycenter norm are called simplex-like.
Point sets with simplex-like margin triples are completely filled in all but the last
layer.
Simplex-like margin triples
The barycenter norm
three margin vectors.
P
q∈Q (qx
+ qy + qz ) of a point set Q depends only on the
If we have given margins with minimal barycenter norm, then to construct Q with
these margins we must place everything as close to the origin as possible (in taxicab
distance).
Margin triples with minimal barycenter norm are called simplex-like.
Point sets with simplex-like margin triples are completely filled in all but the last
layer.
Simplex-like margin triples
The barycenter norm
three margin vectors.
P
q∈Q (qx
+ qy + qz ) of a point set Q depends only on the
If we have given margins with minimal barycenter norm, then to construct Q with
these margins we must place everything as close to the origin as possible (in taxicab
distance).
Margin triples with minimal barycenter norm are called simplex-like.
Point sets with simplex-like margin triples are completely filled in all but the last
layer.
Simplex-like margin triples
The barycenter norm
three margin vectors.
P
q∈Q (qx
+ qy + qz ) of a point set Q depends only on the
If we have given margins with minimal barycenter norm, then to construct Q with
these margins we must place everything as close to the origin as possible (in taxicab
distance).
Margin triples with minimal barycenter norm are called simplex-like.
Point sets with simplex-like margin triples are completely filled in all but the last
layer.
Simplex-like margin triples
The barycenter norm
three margin vectors.
P
q∈Q (qx
+ qy + qz ) of a point set Q depends only on the
If we have given margins with minimal barycenter norm, then to construct Q with
these margins we must place everything as close to the origin as possible (in taxicab
distance).
Margin triples with minimal barycenter norm are called simplex-like.
Point sets with simplex-like margin triples are completely filled in all but the last
layer.
Simplex-like margin triples
The barycenter norm
three margin vectors.
P
q∈Q (qx
+ qy + qz ) of a point set Q depends only on the
If we have given margins with minimal barycenter norm, then to construct Q with
these margins we must place everything as close to the origin as possible (in taxicab
distance).
Margin triples with minimal barycenter norm are called simplex-like.
Point sets with simplex-like margin triples are completely filled in all but the last
layer.
Simplex-like margin triples
The barycenter norm
three margin vectors.
P
q∈Q (qx
+ qy + qz ) of a point set Q depends only on the
If we have given margins with minimal barycenter norm, then to construct Q with
these margins we must place everything as close to the origin as possible (in taxicab
distance).
Margin triples with minimal barycenter norm are called simplex-like.
Point sets with simplex-like margin triples are completely filled in all but the last
layer.
Simplex-like margin triples
The barycenter norm
three margin vectors.
P
q∈Q (qx
+ qy + qz ) of a point set Q depends only on the
If we have given margins with minimal barycenter norm, then to construct Q with
these margins we must place everything as close to the origin as possible (in taxicab
distance).
Margin triples with minimal barycenter norm are called simplex-like.
Point sets with simplex-like margin triples are completely filled in all but the last
layer.
Simplex-like margin triples
The barycenter norm
three margin vectors.
P
q∈Q (qx
+ qy + qz ) of a point set Q depends only on the
If we have given margins with minimal barycenter norm, then to construct Q with
these margins we must place everything as close to the origin as possible (in taxicab
distance).
Margin triples with minimal barycenter norm are called simplex-like.
Point sets with simplex-like margin triples are completely filled in all but the last
layer.
Simplex-like margin triples
The barycenter norm
three margin vectors.
P
q∈Q (qx
+ qy + qz ) of a point set Q depends only on the
If we have given margins with minimal barycenter norm, then to construct Q with
these margins we must place everything as close to the origin as possible (in taxicab
distance).
Margin triples with minimal barycenter norm are called simplex-like.
Point sets with simplex-like margin triples are completely filled in all but the last
layer.
Simplex-like margin triples
The barycenter norm
three margin vectors.
P
q∈Q (qx
+ qy + qz ) of a point set Q depends only on the
If we have given margins with minimal barycenter norm, then to construct Q with
these margins we must place everything as close to the origin as possible (in taxicab
distance).
Margin triples with minimal barycenter norm are called simplex-like.
Point sets with simplex-like margin triples are completely filled in all but the last
layer.
Simplex-like margin triples
The barycenter norm
three margin vectors.
P
q∈Q (qx
+ qy + qz ) of a point set Q depends only on the
If we have given margins with minimal barycenter norm, then to construct Q with
these margins we must place everything as close to the origin as possible (in taxicab
distance).
Margin triples with minimal barycenter norm are called simplex-like.
Point sets with simplex-like margin triples are completely filled in all but the last
layer.
Simplex-like margin triples
The barycenter norm
three margin vectors.
P
q∈Q (qx
+ qy + qz ) of a point set Q depends only on the
If we have given margins with minimal barycenter norm, then to construct Q with
these margins we must place everything as close to the origin as possible (in taxicab
distance).
Margin triples with minimal barycenter norm are called simplex-like.
Point sets with simplex-like margin triples are completely filled in all but the last
layer.
Simplex-like margin triples
The barycenter norm
three margin vectors.
P
q∈Q (qx
+ qy + qz ) of a point set Q depends only on the
If we have given margins with minimal barycenter norm, then to construct Q with
these margins we must place everything as close to the origin as possible (in taxicab
distance).
Margin triples with minimal barycenter norm are called simplex-like.
Point sets with simplex-like margin triples are completely filled in all but the last
layer.
Simplex-like margin triples
The barycenter norm
three margin vectors.
P
q∈Q (qx
+ qy + qz ) of a point set Q depends only on the
If we have given margins with minimal barycenter norm, then to construct Q with
these margins we must place everything as close to the origin as possible (in taxicab
distance).
Margin triples with minimal barycenter norm are called simplex-like.
Point sets with simplex-like margin triples are completely filled in all but the last
layer.
Simplex-like margin triples
The barycenter norm
three margin vectors.
P
q∈Q (qx
+ qy + qz ) of a point set Q depends only on the
If we have given margins with minimal barycenter norm, then to construct Q with
these margins we must place everything as close to the origin as possible (in taxicab
distance).
Margin triples with minimal barycenter norm are called simplex-like.
Point sets with simplex-like margin triples are completely filled in all but the last
layer.
Simplex-like margin triples
The barycenter norm
three margin vectors.
P
q∈Q (qx
+ qy + qz ) of a point set Q depends only on the
If we have given margins with minimal barycenter norm, then to construct Q with
these margins we must place everything as close to the origin as possible (in taxicab
distance).
Margin triples with minimal barycenter norm are called simplex-like.
Point sets with simplex-like margin triples are completely filled in all but the last
layer.
Simplex-like margin triples
The barycenter norm
three margin vectors.
P
q∈Q (qx
+ qy + qz ) of a point set Q depends only on the
If we have given margins with minimal barycenter norm, then to construct Q with
these margins we must place everything as close to the origin as possible (in taxicab
distance).
Margin triples with minimal barycenter norm are called simplex-like.
Point sets with simplex-like margin triples are completely filled in all but the last
layer.
Simplex-like margin triples
The barycenter norm
three margin vectors.
P
q∈Q (qx
+ qy + qz ) of a point set Q depends only on the
If we have given margins with minimal barycenter norm, then to construct Q with
these margins we must place everything as close to the origin as possible (in taxicab
distance).
Margin triples with minimal barycenter norm are called simplex-like.
Point sets with simplex-like margin triples are completely filled in all but the last
layer.
Simplex-like margin triples
The barycenter norm
three margin vectors.
P
q∈Q (qx
+ qy + qz ) of a point set Q depends only on the
If we have given margins with minimal barycenter norm, then to construct Q with
these margins we must place everything as close to the origin as possible (in taxicab
distance).
Margin triples with minimal barycenter norm are called simplex-like.
Point sets with simplex-like margin triples are completely filled in all but the last
layer.
Christian Ikenmeyer
Deciding Positivity of Kronecker Coefficients is NP-hard
7
Simplex-like margins
P
The barycenter norm q∈Q (qx + qy + qz ) of a point set Q depends only on the
three margin vectors.
If we have given margins with minimal barycenter norm, then to construct Q with
these margins we must place everything as close to the origin as possible (in taxicab
distance).
These margin triples are called simplex-like.
Point sets with simplex-like margin triples are completely filled in all but the last
layer.
Caveat
A simplex-like margin triple might not come from an actual
point set Q. There might be margin triples with minimal
barycenter norm that have no corresponding point set Q.
Christian Ikenmeyer
Deciding Positivity of Kronecker Coefficients is NP-hard
8
Simplex-like margins
P
The barycenter norm q∈Q (qx + qy + qz ) of a point set Q depends only on the
three margin vectors.
If we have given margins with minimal barycenter norm, then to construct Q with
these margins we must place everything as close to the origin as possible (in taxicab
distance).
These margin triples are called simplex-like.
Point sets with simplex-like margin triples are completely filled in all but the last
layer.
Caveat
A simplex-like margin triple might not come from an actual
point set Q. There might be margin triples with minimal
barycenter norm that have no corresponding point set Q.
Proposition: All point sets with simplex-like margin triples are pyramids. Therefore
pλ,µ,ν = gλ,µ,ν = tλ,µ,ν .
t
for simplex like triples (λ , µ , ν t ).
Christian Ikenmeyer
t
Deciding Positivity of Kronecker Coefficients is NP-hard
8
Brunetti, Del Lungo, Gerard 2001: Deciding positivity of tλ,µ,ν (= pλ,µ,ν ) for
simplex-like margins (λt , µt , ν t ) is NP-hard.
Christian Ikenmeyer
Deciding Positivity of Kronecker Coefficients is NP-hard
9
Brunetti, Del Lungo, Gerard 2001: Deciding positivity of tλ,µ,ν (= pλ,µ,ν ) for
simplex-like margins (λt , µt , ν t ) is NP-hard.
Since for simplex-like margins we have tλ,µ,ν = gλ,µ,ν , it follows that deciding
positivity of Kronecker coefficients is NP-hard.
Christian Ikenmeyer
Deciding Positivity of Kronecker Coefficients is NP-hard
9
Brunetti, Del Lungo, Gerard 2001: Deciding positivity of tλ,µ,ν (= pλ,µ,ν ) for
simplex-like margins (λt , µt , ν t ) is NP-hard.
Since for simplex-like margins we have tλ,µ,ν = gλ,µ,ν , it follows that deciding
positivity of Kronecker coefficients is NP-hard.
Remark: We used this fact to construct many nontrivially vanishing Kronecker
coefficients, needed in the context of geometric complexity theory.
Christian Ikenmeyer
Deciding Positivity of Kronecker Coefficients is NP-hard
9
Brunetti, Del Lungo, Gerard 2001: Deciding positivity of tλ,µ,ν (= pλ,µ,ν ) for
simplex-like margins (λt , µt , ν t ) is NP-hard.
Since for simplex-like margins we have tλ,µ,ν = gλ,µ,ν , it follows that deciding
positivity of Kronecker coefficients is NP-hard.
Remark: We used this fact to construct many nontrivially vanishing Kronecker
coefficients, needed in the context of geometric complexity theory.
Thank you for your attention.
Christian Ikenmeyer
Deciding Positivity of Kronecker Coefficients is NP-hard
9