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edit approved Nov 10, 2017 at 7:09
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edit approved Nov 10, 2017 at 7:09
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Why are the divided difference operators of the nilHeckenil Hecke ring only of degree 1?

In the paper "A Diagrammatic Approach to the Categorification of Quantum Groups I" (arXiv, journal, MSN), Khovanov and Lauda put a strand with a single dot in degree 2, and put the crossing operator in degree 1, and set these to be the generators of a graded ring. 

However, when they define an action of their (graded) diagram ring on a given (graded) polynomial ring, applying the crossing operator a homogeneous polynomial with integral coefficients in this graded ring yields a polynomial in odd degree, which shouldn't be possible (as attaching dots to strands represents raising the degree by 1, and thus all polynomials sit in even degree). Is there something I'm missing here?

Why are the divided difference operators of the nilHecke ring only of degree 1?

In the paper "A Diagrammatic Approach to the Categorification of Quantum Groups I" (arXiv, journal, MSN), Khovanov and Lauda put a strand with a single dot in degree 2, and put the crossing operator in degree 1, and set these to be the generators of a graded ring. However, when they define an action of their (graded) diagram ring on a given (graded) polynomial ring, applying the crossing operator a homogeneous polynomial with integral coefficients in this graded ring yields a polynomial in odd degree, which shouldn't be possible (as attaching dots to strands represents raising the degree by 1, and thus all polynomials sit in even degree). Is there something I'm missing here?

Why are the divided difference operators of the nil Hecke ring only of degree 1?

In the paper "A Diagrammatic Approach to the Categorification of Quantum Groups I" (arXiv, journal, MSN), Khovanov and Lauda put a strand with a single dot in degree 2, and put the crossing operator in degree 1, and set these to be the generators of a graded ring. 

However, when they define an action of their (graded) diagram ring on a given (graded) polynomial ring, applying the crossing operator a homogeneous polynomial with integral coefficients in this graded ring yields a polynomial in odd degree, which shouldn't be possible (as attaching dots to strands represents raising the degree by 1, and thus all polynomials sit in even degree). Is there something I'm missing here?

Added link to various appearances of paper; removed "Any help would be greatly appreciated")
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LSpice
LSpice
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In the paper "A Diagrammatic Approach to the Categorification of Quantum Groups"Groups I" (arXiv:0803.4121arXiv, journal, MSN), Khovanov and Lauda put a strand with a single dot in degree 2, and put the crossing operator in degree 1, and set these to be the generators of a graded ring. However, when they define an action of their (graded) diagram ring on a given (graded) polynomial ring, applying the crossing operator a homogeneous polynomial with integral coefficients in this graded ring yields a polynomial in odd degree, which shouldn't be possible (as attaching dots to strands represents raising the degree by 1, and thus all polynomials sit in even degree). Is there something I'm missing here? Any help would be greatly appreciated.

In the paper "A Diagrammatic Approach to the Categorification of Quantum Groups" (arXiv:0803.4121), Khovanov and Lauda put a strand with a single dot in degree 2, and put the crossing operator in degree 1, and set these to be the generators of a graded ring. However, when they define an action of their (graded) diagram ring on a given (graded) polynomial ring, applying the crossing operator a homogeneous polynomial with integral coefficients in this graded ring yields a polynomial in odd degree, which shouldn't be possible (as attaching dots to strands represents raising the degree by 1, and thus all polynomials sit in even degree). Is there something I'm missing here? Any help would be greatly appreciated.

In the paper "A Diagrammatic Approach to the Categorification of Quantum Groups I" (arXiv, journal, MSN), Khovanov and Lauda put a strand with a single dot in degree 2, and put the crossing operator in degree 1, and set these to be the generators of a graded ring. However, when they define an action of their (graded) diagram ring on a given (graded) polynomial ring, applying the crossing operator a homogeneous polynomial with integral coefficients in this graded ring yields a polynomial in odd degree, which shouldn't be possible (as attaching dots to strands represents raising the degree by 1, and thus all polynomials sit in even degree). Is there something I'm missing here?

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cantwellnc
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Why are the divided difference operators of the nilHecke ring only of degree 1?

In the paper "A Diagrammatic Approach to the Categorification of Quantum Groups" (arXiv:0803.4121), Khovanov and Lauda put a strand with a single dot in degree 2, and put the crossing operator in degree 1, and set these to be the generators of a graded ring. However, when they define an action of their (graded) diagram ring on a given (graded) polynomial ring, applying the crossing operator a homogeneous polynomial with integral coefficients in this graded ring yields a polynomial in odd degree, which shouldn't be possible (as attaching dots to strands represents raising the degree by 1, and thus all polynomials sit in even degree). Is there something I'm missing here? Any help would be greatly appreciated.