Plücker relations in Sagemath from Macaulay2

Question

I am trying to implement the Plücker relations in Sagemath. Sage has an interface for Macaulay2, and this latter has a command Grassmannian(k-1, n-1) for computing the Plücker relations of the Grassmannian $\text{Gr}(k,n)$.

My problem is: when I do this, Sage returns me a Macaulay2 object, which I want to turn into a Sage object. When I try the command Grassmannian(k-1, n-1).sage(), the compiler returns:

ValueError: variable name 'p_{0' is not alphanumeric.

So I guess the problem is that Macaulay2 uses variables p_{i,j,k}, and this is a problem for Sage. Do you know how can I fix it?

EDIT: here is the code:

R = macaulay2('QQ[apply(subsets(6,3), i -> p_i)]')
G = macaulay2.Grassmannian(2,5,R)
G.sage()

Answer 1

You can use variables that SageMath can understand:

sage: macaulay2('apply(subsets(6,3), i -> "p" | concatenate(apply(i, toString)))')
{p012, p013, p023, p123, p014, p024, p124, p034, p134, p234, p015, p025, p125, p035, p135, p235, p045, p145, p245, p345}

So:

sage: R = macaulay2('QQ[apply(subsets(6,3), i -> "p" | concatenate(apply(i, toString)))]')
sage: G = macaulay2.Grassmannian(2,5,R)
sage: G.sage()
Ideal (p235*p145 - p135*p245 + p125*p345, p234*p145 - p134*p245 + p124*p345, p235*p045 - p035*p245 + p025*p345, p135*p045 - p035*p145 + p015*p345, p125*p045 - p025*p145 + p015*p245, p234*p045 - p034*p245 + p024*p345, p134*p045 - p034*p145 + p014*p345, p124*p045 - p024*p145 + p014*p245, p123*p045 - p023*p145 + p013*p245 - p012*p345, p234*p135 - p134*p235 + p123*p345, p125*p035 - p025*p135 + p015*p235, p234*p035 - p034*p235 + p023*p345, p134*p035 - p034*p135 + p013*p345, p124*p035 - p024*p135 + p014*p235 + p012*p345, p123*p035 - p023*p135 + p013*p235, p234*p125 - p124*p235 + p123*p245, p134*p125 - p124*p135 + p123*p145, p034*p125 - p024*p135 + p014*p235 + p023*p145 - p013*p245 + p012*p345, p234*p025 - p024*p235 + p023*p245, p134*p025 - p024*p135 + p023*p145 + p012*p345, p034*p025 - p024*p035 + p023*p045, p124*p025 - p024*p125 + p012*p245, p123*p025 - p023*p125 + p012*p235, p234*p015 - p014*p235 + p013*p245 - p012*p345, p134*p015 - p014*p135 + p013*p145, p034*p015 - p014*p035 + p013*p045, p124*p015 - p014*p125 + p012*p145, p024*p015 - p014*p025 + p012*p045, p123*p015 - p013*p125 + p012*p135, p023*p015 - p013*p025 + p012*p035, p124*p034 - p024*p134 + p014*p234, p123*p034 - p023*p134 + p013*p234, p123*p024 - p023*p124 + p012*p234, p123*p014 - p013*p124 + p012*p134, p023*p014 - p013*p024 + p012*p034) of Multivariate Polynomial Ring in p012, p013, p023, p123, p014, p024, p124, p034, p134, p234, p015, p025, p125, p035, p135, p235, p045, p145, p245, p345 over Rational Field

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When you have numbers higher than 9 appearing you can add a separator like this:

sage: macaulay2('apply(subsets(11,11), i -> "p" | concatenate(apply(i, v -> "x" | toString v)))')
{px0x1x2x3x4x5x6x7x8x9x10}

I tried using an underscore as the separator at first but I didn't manage to make it work.