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 SteenrodReduce

Source Notebook

Evaluate products and sums of monomials in the Steenrod algebra for the prime 2 written in the Serre–Cartan basis

Contributed by: Elliot Granath

ResourceFunction["SteenrodReduce"][{list}]

reduces list={i1,i2,…,ir} corresponding to monomial in the Steenrod algebra via the Adem relations.

ResourceFunction["SteenrodReduce"][{list1,list2,…,listn}]

reduces each listi={j1,j2,…,jri} corresponding to and outputs the sum as a list of lists.

Details and Options

A monomial Sqi1Sqi2⋯Sqir in the Serre-Cartan basis is expressed here as the sequence {i1,i2,…,ir}. Multiple sequences in a list are interpreted as the terms of a polynomial. For example, Sq5Sq1+Sq6 (or, equivalently, Sq5,1+Sq6) is written {{5,1},{6}}.
After the Adem relation reduction, the resulting sum of monomials is given in the output as list of lists.
Zero is written as the null sequence {} to avoid confusion with {0}, which is interpreted as Sq0=1.
The input should always be written as a list of lists, even when the outermost list has length 1.
Outputs are given using the same list notation as inputs.
The input and output always use the Serre-Cartan basis for the Steenrod algebra over ℤ2. Monomials in this basis correspond to admissible sequences {i1,i2,…,ir}, where ij≥2ij+1 for all j<r.

Examples

Basic Examples (3) 

Simplify a product of Steenrod squares as in the computation Sq1Sq2=Sq3:

In[1]:=
ResourceFunction["SteenrodReduce"][{{1, 2}}]
Out[1]=

Show a grid of simplifications for all {a,b} between 1 and 5:

In[2]:=
(* Evaluate this cell to get the example input *) CloudGet["https://www.wolframcloud.com/obj/5e1f12e9-e772-41ca-a695-dd0f616ed41c"]
Out[2]=

Expand Sq32Sq32Sq32 using the Adem relations:

In[3]:=
ResourceFunction["SteenrodReduce"][{{32, 32, 32}}]
Out[3]=

Scope (3) 

Simplify a sum of monomials as in the computation Sq1Sq4+Sq5=0:

In[4]:=
ResourceFunction["SteenrodReduce"][{{1, 4}, {5}}]
Out[4]=

Verify a known equality such as Sq2Sq2Sq2=Sq2Sq1Sq2Sq1:

In[5]:=
ResourceFunction["SteenrodReduce"][{{2, 2, 2}, {2, 1, 2, 1}}]
Out[5]=

Adem relations are applied exhaustively until no further simplifications can be made:

In[6]:=
ResourceFunction["SteenrodReduce"][{{33, 50, 21, 2, 8}}]
Out[6]=

Options (3) 

Specify the output format with OutputForm→"Form":

In[7]:=
ResourceFunction["SteenrodReduce"][{{4, 4, 4}}, OutputForm -> "List"]
Out[7]=

Display the output as a polynomial in Steenrod squares:

In[8]:=
ResourceFunction["SteenrodReduce"][{{4, 4, 4}}, OutputForm -> "Display"]
Out[8]=

Obtain output which can be copied as plaintext into a Tex document:

In[9]:=
ResourceFunction["SteenrodReduce"][{{4, 4, 4}}, OutputForm -> "Tex"]
Out[9]=

Publisher

Elliot Granath

Requirements

Wolfram Language 13.0 (December 2021) or above

Version History

  • 1.0.0 – 07 February 2025

License Information